Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.6 AIME Find the answer

3. In the convex quadrilateral $ABCD$ , $\angle ADB=70^{\circ}, \angle CDB=$ $40^{\circ}, \angle BAC=20^{\circ}, \angle BCA=35^{\circ}$ . Find the degree measure of the acute angle between $AC$ and $BD$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

(提示: Extend $B D$ to point $P$ , such that $D P=D B$ . It is only necessary to prove that points $A$ , $B$ , $C$ , and $P$ are concyclic, then $D C=D P=D A, \angle A D C=110^{\circ}$ , $\angle A C D=35^{\circ}$ . Therefore, the included acute angle is $75^{\circ}$ .)

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.