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thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Circles intersecting with triangles

Let \overline{PQ} be a diameter of a circle and T be a point on the circle besides P and Q. The tangent to the circle through point Q intersects line PT at R, and the tangent through T intersects \overline{QR} at M. Prove that M is the midpoint of \overline{QR}.
thanks!

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Good. You can read.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: Circles intersecting with triangles

hi

TQR =  TPQ (angle made by tangent with chord = angle subtended by chord)

So you should be able to show that triangles PTQ and QTR are similar.

TM = QM (as tangents from the same point are equal)

So extend the similarity property to include the midpoints of PQ and QR.

Bob

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