MathLabs

Problem 6

Two planes PP and QQ intersect along line pp. Points A∈PA\in P and C∈QC\in Q are not on pp. Construct an isosceles trapezoid ABCDABCD with AB∥DCAB\parallel DC, an incircle, and vertices B∈PB\in P, D∈QD\in Q.
Step 2 of 5: Use the tangential trapezoid condition
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AB+DC=AD+BC=2AD⟹AD=AB+DC2AB+DC=AD+BC=2AD\quad\Longrightarrow\quad AD=\frac{AB+DC}{2}
Detailed analysis

For a quadrilateral with an incircle, opposite-side sums are equal: AB+DC=AD+BCAB+DC=AD+BC. Since the trapezoid is isosceles, AD=BCAD=BC, so each leg equals (AB+DC)/2(AB+DC)/2.