MathLabs

Problem 1

In a convex quadrilateral of area 3232, the sum of the lengths of two opposite sides and one diagonal is 1616. Determine all possible lengths of the other diagonal.
Step 3 of 7: Bound the altitudes
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In plain words

Equality will later identify the extremal shape.

d1≤AB,d2≤DCd_1\le AB,\quad d_2\le DC
Detailed analysis

A perpendicular distance is no larger than the distance to any point on the line, so d1≤ABd_1\le AB and d2≤DCd_2\le DC. Equality requires AB⊥BDAB\perp BD and DC⊥BDDC\perp BD.