Download Problem #522 Solution In the diagram below, AB = AC and BY = CZ

Problem #522
Solution
In the diagram below, AB = AC and BY = CZ. Let P the intersection
of segments BY and CZ. Is it necessarily true that P Y = P Z? Prove it,
or else give a counterexample diagram where all of the hypotheses hold but
P Y and P Z clearly have different lengths.
A
Z
Y
M
Y0
P
P0
B
C
Answer: It is not necessarily true that P Y = P Z.
In the diagram above, M is the foot of the perpendicular from B to AC,
and BY 0 is the reflection of BY around BM . Triangles BM Y and BM Y 0
are congruent by ASA, so BY 0 = BY . On the other hand, P 0 Y 0 6= P 0 Z.
Source: College Geometry by Martin Isaacs