Problem
For a quadrilateral $ABCD$, if the bisectors of $∠A$ and $∠C$ intersect on the diagonal $BD$, then the bisectors of $∠B$ and $∠D$ intersect on the diagonal $AC$.
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Solution
If the intersection point of the bisector of $∠A$ and $∠C$ is $P$ and it is on the diagonal $BD$, then from the angle bisector theorem,
$$AB∶AD=BP∶DP=CB∶CD,$$
$$AB×CD=AD×CB,$$
$$\frac{AB}{CB}=\frac{AD}{CD},$$
$$∴ \ AB∶CB=AD∶CD. \qquad [1]$$
If the intersection point of the bisector of $∠B$ and $AC$ is $Q$,
$$AB∶CB=AQ∶CQ. \qquad [2]$$
From $[1]$ and $[2]$,
$$AD∶CD=AQ∶CQ.$$
Therefore, the line $DQ$ is the bisector of $∠D$.
In other words, the bisector of $∠B$ and $∠D$ intersect at $Q$ on the diagonal $AC$.
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Reference Teiichiro Sasabe (1976) The Encyclopedia of Geometry (2nd edition), Seikyo-Shinsha, p.42.