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Problem 2

In isosceles triangle ABCABC with AB=ACAB=AC, let MM be the midpoint of BCBC and let O∈AMO\in AM satisfy OB⊥ABOB\perp AB. For Q∈BCQ\in BC, let E∈ABE\in AB and F∈ACF\in AC be distinct collinear points with QQ. Prove OQ⊥EFOQ\perp EF if and only if QE=QFQE=QF.
Step 1 of 4: Use perpendicularity to get cyclic quadrilaterals
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In plain words

Use perpendicularity to get cyclic quadrilaterals

EBOQ and OQCF are cyclicEBOQ\text{ and }OQCF\text{ are cyclic}
Detailed analysis

Assume OQ⊥EFOQ\perp EF. Since EB⊂ABEB\subset AB and OB⊥ABOB\perp AB, we have ∠EBO=90∘\angle EBO=90^\circ; also EQ⊂EFEQ\subset EF gives ∠EQO=90∘\angle EQO=90^\circ. Hence EBOQEBOQ is cyclic. Similarly, symmetry gives OC⊥ACOC\perp AC, so ∠OCF=∠OQF=90∘\angle OCF=\angle OQF=90^\circ and OQCFOQCF is cyclic.