Let ABC be an isosceles triangle, where AB AC. Suppose that M is a point on the side AB such that AB- MB = AM2. Suppose also that AM = BC. Prove that the angle ZABC is twice ZBAC. [Hint: Consider the circumcircle O of AMC. Show that BC is tangent to O. Use this to show that ZCMB = ZCBM . You are encouraged to use propositions from Books 1, 2, & 3 from Euclid's Elements.] Draw a picture to help support your proof.

Question
Let ABC be an isosceles triangle, where AB AC. Suppose that M is a point
on the side
AB such that AB- MB = AM2. Suppose also that AM = BC. Prove that the angle ZABC is
twice ZBAC. [Hint: Consider the circumcircle O of AMC. Show that BC is tangent to O.
Use this to show that ZCMB = ZCBM . You are encouraged to use
propositions from Books 1,
2, & 3 from Euclid's Elements.] Draw a picture to help support your proof.

Image Transcription

Let ABC be an isosceles triangle, where AB AC. Suppose that M is a point on the side AB such that AB- MB = AM2. Suppose also that AM = BC. Prove that the angle ZABC is twice ZBAC. [Hint: Consider the circumcircle O of AMC. Show that BC is tangent to O. Use this to show that ZCMB = ZCBM . You are encouraged to use propositions from Books 1, 2, & 3 from Euclid's Elements.] Draw a picture to help support your proof.

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