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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.

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.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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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.
Transcribed Image Text: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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