1
00:00:00,690 --> 00:00:03,450
What I want to do in this video
is to prove one of the more

2
00:00:03,450 --> 00:00:08,980
useful results in geometry, and
that's that an inscribed angle

3
00:00:08,980 --> 00:00:14,950
is just an angle whose vertex
sits on the circumference

4
00:00:14,950 --> 00:00:17,080
of the circle.

5
00:00:17,080 --> 00:00:19,800
So that is our inscribed angle.

6
00:00:19,800 --> 00:00:24,950
I'll denote it by psi -- I'll
use the psi for inscribed angle

7
00:00:24,950 --> 00:00:27,170
and angles in this video.

8
00:00:27,170 --> 00:00:33,530
That psi, the inscribed angle,
is going to be exactly 1/2 of

9
00:00:33,530 --> 00:00:37,880
the central angle that
subtends the same arc.

10
00:00:37,880 --> 00:00:40,730
So I just used a lot a fancy
words, but I think you'll

11
00:00:40,730 --> 00:00:41,650
get what I'm saying.

12
00:00:41,650 --> 00:00:42,820
So this is psi.

13
00:00:42,820 --> 00:00:44,470
It is an inscribed angle.

14
00:00:44,470 --> 00:00:48,710
It sits, its vertex sits
on the circumference.

15
00:00:48,710 --> 00:00:52,570
And if you draw out the two rays
that come out from this angle

16
00:00:52,570 --> 00:00:56,040
or the two cords that define
this angle, it intersects the

17
00:00:56,040 --> 00:00:57,340
circle at the other end.

18
00:00:57,340 --> 00:01:00,390
And if you look at the part of
the circumference of the circle

19
00:01:00,390 --> 00:01:03,730
that's inside of it, that
is the arc that is

20
00:01:03,730 --> 00:01:06,160
subtended by psi.

21
00:01:06,160 --> 00:01:09,010
It's all very fancy words,
but I think the idea is

22
00:01:09,010 --> 00:01:09,920
pretty straightforward.

23
00:01:09,920 --> 00:01:28,485
This right here is the arc
subtended by psi, where psi is

24
00:01:28,485 --> 00:01:31,560
that inscribed angle right over
there, the vertex sitting

25
00:01:31,560 --> 00:01:32,400
on the circumference.

26
00:01:32,400 --> 00:01:37,920
Now, a central angle is an
angle where the vertex is

27
00:01:37,920 --> 00:01:39,460
sitting at the center
of the circle.

28
00:01:39,460 --> 00:01:41,880
So let's say that this right
here -- I'll try to eyeball

29
00:01:41,880 --> 00:01:45,510
it -- that right there is
the center of the circle.

30
00:01:45,510 --> 00:01:51,360
So let me draw a central angle
that subtends this same arc.

31
00:01:51,360 --> 00:01:58,470
So that looks like a central
angle subtending that same arc.

32
00:01:58,470 --> 00:01:59,390
Just like that.

33
00:01:59,390 --> 00:02:01,440
Let's call this theta.

34
00:02:01,440 --> 00:02:06,030
So this angle is psi, this
angle right here is theta.

35
00:02:06,030 --> 00:02:10,120
What I'm going to prove in this
video is that psi is always

36
00:02:10,120 --> 00:02:14,050
going to be equal
to 1/2 of theta.

37
00:02:14,050 --> 00:02:18,220
So if I were to tell you that
psi is equal to, I don't know,

38
00:02:18,220 --> 00:02:21,330
25 degrees, then you would
immediately know that theta

39
00:02:21,330 --> 00:02:23,090
must be equal to 50 degrees.

40
00:02:23,090 --> 00:02:26,080
Or if I told you that theta was
80 degrees, then you would

41
00:02:26,080 --> 00:02:29,300
immediately know that
psi was 40 degrees.

42
00:02:29,300 --> 00:02:31,500
So let's actually proved this.

43
00:02:31,500 --> 00:02:34,520
So let me clear this.

44
00:02:34,520 --> 00:02:37,730
So a good place to start,
or the place I'm going to

45
00:02:37,730 --> 00:02:40,460
start, is a special case.

46
00:02:40,460 --> 00:02:45,250
I'm going to draw an inscribed
angle, but one of the chords

47
00:02:45,250 --> 00:02:47,910
that define it is going to be
the diameter of the circle.

48
00:02:47,910 --> 00:02:50,526
So this isn't going to be the
general case, this is going

49
00:02:50,526 --> 00:02:51,320
to be a special case.

50
00:02:51,320 --> 00:02:55,325
So let me see, this is the
center right here of my circle.

51
00:02:55,325 --> 00:02:59,030
I'm trying to eyeball it.

52
00:02:59,030 --> 00:03:00,770
Center looks like that.

53
00:03:00,770 --> 00:03:04,210
So let me draw a diameter.

54
00:03:04,210 --> 00:03:06,440
So the diameter
looks like that.

55
00:03:06,440 --> 00:03:09,410
Then let me define
my inscribed angle.

56
00:03:09,410 --> 00:03:11,860
This diameter is
one side of it.

57
00:03:11,860 --> 00:03:15,910
And then the other side
maybe is just like that.

58
00:03:15,910 --> 00:03:20,520
So let me call this
right here psi.

59
00:03:20,520 --> 00:03:27,120
If that's psi, this length right
here is a radius -- that's

60
00:03:27,120 --> 00:03:29,330
our radius of our circle.

61
00:03:29,330 --> 00:03:33,080
Then this length right here is
also going to be the radius of

62
00:03:33,080 --> 00:03:35,760
our circle going from the
center to the circumference.

63
00:03:35,760 --> 00:03:38,130
Your circumference is defined
by all of the points that are

64
00:03:38,130 --> 00:03:40,340
exactly a radius away
from the center.

65
00:03:40,340 --> 00:03:43,610
So that's also a radius.

66
00:03:43,610 --> 00:03:47,920
Now, this triangle right here
is an isosceles triangle.

67
00:03:47,920 --> 00:03:49,890
It has two sides
that are equal.

68
00:03:49,890 --> 00:03:51,880
Two sides that are
definitely equal.

69
00:03:51,880 --> 00:03:54,630
We know that when we have two
sides being equal, their

70
00:03:54,630 --> 00:03:57,290
base angles are also equal.

71
00:03:57,290 --> 00:04:00,640
So this will also
be equal to psi.

72
00:04:00,640 --> 00:04:02,130
You might not recognize
it because it's

73
00:04:02,130 --> 00:04:03,180
tilted up like that.

74
00:04:03,180 --> 00:04:05,720
But I think many of us when we
see a triangle that looks like

75
00:04:05,720 --> 00:04:10,940
this, if I told you this is r
and that is r, that these two

76
00:04:10,940 --> 00:04:17,860
sides are equal, and if this is
psi, then you would also

77
00:04:17,860 --> 00:04:20,830
know that this angle is
also going to be psi.

78
00:04:20,830 --> 00:04:23,930
Base angles are equivalent
on an isosceles triangle.

79
00:04:23,930 --> 00:04:26,720
So this is psi, that is also psi.

80
00:04:26,720 --> 00:04:29,770
Now, let me look at
the central angle.

81
00:04:29,770 --> 00:04:32,710
This is the central angle
subtending the same arc.

82
00:04:32,710 --> 00:04:35,920
Let's highlight the arc that
they're both subtending.

83
00:04:35,920 --> 00:04:40,300
This right here is the arc that
they're both going to subtend.

84
00:04:40,300 --> 00:04:44,350
So this is my central
angle right there, theta.

85
00:04:44,350 --> 00:04:49,000
Now if this angle is theta,
what's this angle going to be?

86
00:04:49,000 --> 00:04:50,620
This angle right here.

87
00:04:50,620 --> 00:04:53,010
Well, this angle is
supplementary to theta,

88
00:04:53,010 --> 00:04:56,640
so it's 180 minus theta.

89
00:04:56,640 --> 00:04:59,560
When you add these two angles
together you go 180 degrees

90
00:04:59,560 --> 00:05:01,750
around or they kind
of form a line.

91
00:05:01,750 --> 00:05:03,790
They're supplementary
to each other.

92
00:05:03,790 --> 00:05:06,740
Now we also know that these
three angles are sitting

93
00:05:06,740 --> 00:05:08,260
inside of the same triangle.

94
00:05:08,260 --> 00:05:12,030
So they must add up
to 180 degrees.

95
00:05:12,030 --> 00:05:19,300
So we get psi -- this psi plus
that psi plus psi plus this

96
00:05:19,300 --> 00:05:25,420
angle, which is 180 minus
theta plus 180 minus theta.

97
00:05:25,420 --> 00:05:29,130
These three angles must
add up to 180 degrees.

98
00:05:29,130 --> 00:05:31,740
They're the three
angles of a triangle.

99
00:05:31,740 --> 00:05:34,605
Now we could subtract
180 from both sides.

100
00:05:37,140 --> 00:05:43,260
psi plus psi is 2 psi minus
theta is equal to 0.

101
00:05:43,260 --> 00:05:44,840
Add theta to both sides.

102
00:05:44,840 --> 00:05:48,770
You get 2 psi is equal to theta.

103
00:05:48,770 --> 00:05:52,850
Multiply both sides by 1/2
or divide both sides by 2.

104
00:05:52,850 --> 00:05:56,680
You get psi is equal
to 1/2 of theta.

105
00:05:56,680 --> 00:06:00,070
So we just proved what we set
out to prove for the special

106
00:06:00,070 --> 00:06:07,120
case where our inscribed angle
is defined, where one of the

107
00:06:07,120 --> 00:06:11,200
rays, if you want to view these
lines as rays, where one of the

108
00:06:11,200 --> 00:06:15,220
rays that defines this
inscribed angle is

109
00:06:15,220 --> 00:06:17,180
along the diameter.

110
00:06:17,180 --> 00:06:19,200
The diameter forms
part of that ray.

111
00:06:19,200 --> 00:06:21,720
So this is a special
case where one edge is

112
00:06:21,720 --> 00:06:23,760
sitting on the diameter.

113
00:06:23,760 --> 00:06:27,660
So already we could
generalize this.

114
00:06:27,660 --> 00:06:30,580
So now that we know that if
this is 50 that this is

115
00:06:30,580 --> 00:06:32,820
going to be 100 degrees
and likewise, right?

116
00:06:32,820 --> 00:06:37,460
Whatever psi is or whatever
theta is, psi's going to be 1/2

117
00:06:37,460 --> 00:06:40,450
of that, or whatever psi is,
theta is going to

118
00:06:40,450 --> 00:06:41,830
be 2 times that.

119
00:06:41,830 --> 00:06:44,110
And now this will
apply for any time.

120
00:06:44,110 --> 00:06:55,440
We could use this notion any
time that -- so just using that

121
00:06:55,440 --> 00:06:59,460
result we just got, we can now
generalize it a little bit,

122
00:06:59,460 --> 00:07:02,890
although this won't apply
to all inscribed angles.

123
00:07:02,890 --> 00:07:05,090
Let's have an inscribed
angle that looks like this.

124
00:07:10,680 --> 00:07:12,980
So this situation, the center,
you can kind of view it as

125
00:07:12,980 --> 00:07:15,470
it's inside of the angle.

126
00:07:15,470 --> 00:07:17,150
That's my inscribed angle.

127
00:07:17,150 --> 00:07:18,890
And I want to find a
relationship between this

128
00:07:18,890 --> 00:07:22,450
inscribed angle and the central
angle that's subtending

129
00:07:22,450 --> 00:07:24,360
to same arc.

130
00:07:24,360 --> 00:07:29,880
So that's my central angle
subtending the same arc.

131
00:07:29,880 --> 00:07:33,550
Well, you might say, hey, gee,
none of these ends or these

132
00:07:33,550 --> 00:07:37,310
chords that define this angle,
neither of these are diameters,

133
00:07:37,310 --> 00:07:40,400
but what we can do is
we can draw a diameter.

134
00:07:40,400 --> 00:07:43,300
If the center is within
these two chords we

135
00:07:43,300 --> 00:07:46,100
can draw a diameter.

136
00:07:46,100 --> 00:07:48,920
We can draw a diameter
just like that.

137
00:07:48,920 --> 00:07:51,680
If we draw a diameter just like
that, if we define this angle

138
00:07:51,680 --> 00:07:55,430
as psi 1, that angle as psi 2.

139
00:07:55,430 --> 00:07:58,320
Clearly psi is the sum
of those two angles.

140
00:07:58,320 --> 00:08:04,350
And we call this angle theta
1, and this angle theta 2.

141
00:08:04,350 --> 00:08:07,240
We immediately you know that,
just using the result I just

142
00:08:07,240 --> 00:08:12,540
got, since we have one side of
our angles in both cases being

143
00:08:12,540 --> 00:08:18,260
a diameter now, we know
that psi 1 is going to be

144
00:08:18,260 --> 00:08:22,010
equal to 1/2 theta 1.

145
00:08:22,010 --> 00:08:24,870
And we know that psi 2 is
going to be 1/2 theta 2.

146
00:08:24,870 --> 00:08:30,140
Psi 2 is going to
be 1/2 theta 2.

147
00:08:30,140 --> 00:08:39,850
So psi, which is psi 1 plus psi 2,
so psi 1 plus psi 2 is going to

148
00:08:39,850 --> 00:08:41,120
be equal to these two things.

149
00:08:41,120 --> 00:08:47,580
1/2 theta 1 plus 1/2 theta 2.

150
00:08:47,580 --> 00:08:51,180
psi 1 plus psi 2, this is equal
to the first inscribed

151
00:08:51,180 --> 00:08:53,850
angle that we want to deal
with, just regular psi.

152
00:08:53,850 --> 00:08:54,980
That's psi.

153
00:08:54,980 --> 00:08:58,350
And this right here, this
is equal to 1/2 times

154
00:08:58,350 --> 00:09:00,960
theta 1 plus theta 2.

155
00:09:00,960 --> 00:09:03,960
What's theta 1 plus theta 2?

156
00:09:03,960 --> 00:09:06,470
Well that's just our
original theta that

157
00:09:06,470 --> 00:09:08,490
we were dealing with.

158
00:09:08,490 --> 00:09:12,080
So now we see that psi
is equal to 1/2 theta.

159
00:09:12,080 --> 00:09:14,710
So now we've proved it for a
slightly more general case

160
00:09:14,710 --> 00:09:20,020
where our center is inside
of the two rays that

161
00:09:20,020 --> 00:09:21,640
define that angle.

162
00:09:21,640 --> 00:09:27,100
Now, we still haven't addressed
a slightly harder situation or

163
00:09:27,100 --> 00:09:33,660
a more general situation where
if this is the center of our

164
00:09:33,660 --> 00:09:39,420
circle and I have an inscribed
angle where the center isn't

165
00:09:39,420 --> 00:09:40,990
sitting inside of
the two chords.

166
00:09:40,990 --> 00:09:41,820
Let me draw that.

167
00:09:41,820 --> 00:09:48,800
So that's going to be my
vertex, and I'll switch colors,

168
00:09:48,800 --> 00:09:51,540
so let's say that is one of the
chords that defines the

169
00:09:51,540 --> 00:09:53,320
angle, just like that.

170
00:09:53,320 --> 00:09:57,860
And let's say that is the
other chord that defines

171
00:09:57,860 --> 00:09:59,170
the angle just like that.

172
00:09:59,170 --> 00:10:02,500
So how do we find the
relationship between, let's

173
00:10:02,500 --> 00:10:07,910
call, this angle right
here, let's call it psi 1.

174
00:10:07,910 --> 00:10:13,050
How do we find the relationship
between psi 1 and the central

175
00:10:13,050 --> 00:10:16,160
angle that subtends
this same arc?

176
00:10:16,160 --> 00:10:19,530
So when I talk about the same
arc, that's that right there.

177
00:10:19,530 --> 00:10:22,720
So the central angle that
subtends the same arc

178
00:10:22,720 --> 00:10:23,660
will look like this.

179
00:10:28,150 --> 00:10:32,910
Let's call that theta 1.

180
00:10:32,910 --> 00:10:36,770
What we can do is use what we
just learned when one side of

181
00:10:36,770 --> 00:10:39,350
our inscribed angle
is a diameter.

182
00:10:39,350 --> 00:10:41,135
So let's construct that.

183
00:10:41,135 --> 00:10:44,260
So let me draw a diameter here.

184
00:10:44,260 --> 00:10:47,010
The result we want still is
that this should be 1/2 of

185
00:10:47,010 --> 00:10:48,180
this, but let's prove it.

186
00:10:48,180 --> 00:10:57,560
Let's draw a diameter
just like that.

187
00:10:57,560 --> 00:11:09,490
Let me call this angle right
here, let me call that psi 2.

188
00:11:09,490 --> 00:11:14,770
And it is subtending this arc
right there -- let me do

189
00:11:14,770 --> 00:11:16,140
that in a darker color.

190
00:11:16,140 --> 00:11:19,770
It is subtending this
arc right there.

191
00:11:19,770 --> 00:11:22,360
So the central angle that
subtends that same arc,

192
00:11:22,360 --> 00:11:25,300
let me call that theta 2.

193
00:11:25,300 --> 00:11:30,890
Now, we know from the earlier
part of this video that psi

194
00:11:30,890 --> 00:11:37,600
2 is going to be equal
to 1/2 theta 2, right?

195
00:11:37,600 --> 00:11:40,760
They share -- the
diameter is right there.

196
00:11:40,760 --> 00:11:44,300
The diameter is one of the
chords that forms the angle.

197
00:11:44,300 --> 00:11:47,500
So psi 2 is going to be
equal to 1/2 theta 2.

198
00:11:50,140 --> 00:11:52,810
This is exactly what we've been
doing in the last video, right?

199
00:11:52,810 --> 00:11:55,430
This is an inscribed angle.

200
00:11:55,430 --> 00:11:59,550
One of the chords that define
is sitting on the diameter.

201
00:11:59,550 --> 00:12:02,740
So this is going to be 1/2 of
this angle, of the central

202
00:12:02,740 --> 00:12:05,980
angle that subtends
the same arc.

203
00:12:05,980 --> 00:12:09,000
Now, let's look at
this larger angle.

204
00:12:09,000 --> 00:12:11,680
This larger angle right here.

205
00:12:11,680 --> 00:12:14,240
Psi 1 plus psi 2.

206
00:12:14,240 --> 00:12:22,720
Right, that larger angle
is psi 1 plus psi 2.

207
00:12:22,720 --> 00:12:28,680
Once again, this subtends this
entire arc right here, and it

208
00:12:28,680 --> 00:12:32,100
has a diameter as one of the
chords that defines

209
00:12:32,100 --> 00:12:34,310
this huge angle.

210
00:12:34,310 --> 00:12:37,380
So this is going to be 1/2
of the central angle that

211
00:12:37,380 --> 00:12:38,580
subtends the same arc.

212
00:12:38,580 --> 00:12:42,270
We're just using what we've
already shown in this video.

213
00:12:42,270 --> 00:12:47,390
So this is going to be equal to
1/2 of this huge central angle

214
00:12:47,390 --> 00:12:51,370
of theta 1 plus theta 2.

215
00:12:54,310 --> 00:12:56,530
So far we've just used
everything that we've learned

216
00:12:56,530 --> 00:12:58,160
earlier in this video.

217
00:12:58,160 --> 00:13:03,160
Now, we already know that psi
2 is equal to 1/2 theta 2.

218
00:13:03,160 --> 00:13:05,630
So let me make that
substitution.

219
00:13:05,630 --> 00:13:07,030
This is equal to that.

220
00:13:07,030 --> 00:13:15,330
So we can say that si 1 plus
-- instead of si 2 I'll write

221
00:13:15,330 --> 00:13:26,630
1/2 theta 2 is equal to 1/2
theta 1 plus 1/2 theta 2.

222
00:13:30,340 --> 00:13:34,020
We can subtract 1/2 theta
2 from both sides, and

223
00:13:34,020 --> 00:13:35,740
we get our result.

224
00:13:35,740 --> 00:13:40,900
Si 1 is equal to 1/2 theta one.

225
00:13:40,900 --> 00:13:41,970
And now we're done.

226
00:13:41,970 --> 00:13:44,990
We have proven the situation
that the inscribed angle is

227
00:13:44,990 --> 00:13:50,680
always 1/2 of the central angle
that subtends the same arc,

228
00:13:50,680 --> 00:13:53,980
regardless of whether the
center of the circle is inside

229
00:13:53,980 --> 00:13:58,990
of the angle, outside of the
angle, whether we have a

230
00:13:58,990 --> 00:14:00,950
diameter on one side.

231
00:14:00,950 --> 00:14:05,860
So any other angle can be
constructed as a sum of

232
00:14:05,860 --> 00:14:08,300
any or all of these that
we've already done.

233
00:14:08,300 --> 00:14:10,190
So hopefully you found this
useful and now we can actually

234
00:14:10,190 --> 00:14:14,630
build on this result to do some
more interesting

235
00:14:14,630 --> 00:14:16,460
geometry proofs.