Proving the converse to the tangent secant theorem

Watch this thread
username1426065
Badges: 11
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#1
Report Thread starter 6 years ago
#1
Let M be a point not on the circle and let MC be a line intersecting the circle at C.
Let MA be a secant to the circle, intersecting the circle at B and A.
Let |MC|2 = |MA||MB|.
Prove that MC is tangent to the circle at C.

How do I do this? I know that if angle MCB is congruent to angle CAB then that shows that MC is tangent to the circle by another theorem, but I don't know how to show that congruence.
0
reply
Gregorius
Badges: 14
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#2
Report 6 years ago
#2
(Original post by UpstairsMuffin)
Let M be a point not on the circle and let MC be a line intersecting the circle at C.
Let MA be a secant to the circle, intersecting the circle at B and A.
Let |MC|2 = |MA||MB|.
Prove that MC is tangent to the circle at C.

How do I do this? I know that if angle MCB is congruent to angle CAB then that shows that MC is tangent to the circle by another theorem, but I don't know how to show that congruence.
Let F be the centre of the circle, and construct MF. Let E be such that ME is tangent to the circle (put it on the other side for clarity). Apply the tangent secant theorem to ME and MBA. Now start recognizing congruences.
1
reply
Renzhi10122
Badges: 11
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#3
Report 6 years ago
#3
(Original post by UpstairsMuffin)
Let M be a point not on the circle and let MC be a line intersecting the circle at C.
Let MA be a secant to the circle, intersecting the circle at B and A.
Let |MC|2 = |MA||MB|.
Prove that MC is tangent to the circle at C.

How do I do this? I know that if angle MCB is congruent to angle CAB then that shows that MC is tangent to the circle by another theorem, but I don't know how to show that congruence.
Well, we know the converse is true (alternate segment theorem does the job) ie, if MD is tangent, then MD^2=MAMB. Consider now the circle with radius MD center M. This intersects the circle at most 2 times, and by symmetry, it intersects the circle at D' such that MD' is also tangent. Hence, at any other point P on the circle, MP=/= MD unless P=D or D'
0
reply
username1426065
Badges: 11
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#4
Report Thread starter 6 years ago
#4
I'd like to use the theorem that if angle MCB is congruent to angle CAB then MC is tangent to circle ABC, though.
Is there some way I could use similarity?
0
reply
username1426065
Badges: 11
Rep:
? You'll earn badges for being active around the site. Rep gems come when your posts are rated by other community members.
#5
Report Thread starter 6 years ago
#5
Eh alright, I'll just use the proofwiki proof. Thanks.
0
reply
X

Quick Reply

Attached files
Write a reply...
Reply
new posts
Back
to top
Latest

How did The Student Room help you with your university application?

Talking to current university students (17)
18.48%
Talking to peers going through the same thing (31)
33.7%
Speaking to student ambassadors from the universities (5)
5.43%
Speaking to staff members from universities (2)
2.17%
Using the personal statement builder, library or helper service (9)
9.78%
Reading articles about what steps to take (18)
19.57%
Learning about/speaking to Student Finance England (4)
4.35%
Something else (tell us in the thread) (6)
6.52%

Watched Threads

View All