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1/BD - 1/BC =1/2AB if <bAC=90^o, BC - BD = 2AB, BD bisectsd <ABC

Source: 2023 NZMO - New Zealand Maths Olympiad Round 1 p6

September 2, 2023
geometry

Problem Statement

Let triangle ABCABC be right-angled at AA. Let DD be the point on ACAC such that BDBD bisects angle ∠ABC\angle ABC. Prove that BC−BD=2ABBC - BD = 2AB if and only if 1BD−1BC=12AB\frac{1}{BD} - \frac{1}{BC} =\frac{1}{2AB}.
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