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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 3 of 4: Set up the converse line
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In plain words

Set up the converse line

QE=QFQE=QF
Detailed analysis

Assume QE=QFQE=QF. Let E′F′E'F' be the line through QQ perpendicular to OQOQ, with E′∈ABE'\in AB and F′∈ACF'\in AC. The forward implication already proved gives QE′=QF′QE'=QF'.