Problem on triangle similarity.

Question

PQRS is a square. The bisector of angle SQR meets PR and SR at T and V respectively. Prove that: QV*TR = QT*SV

Answer 1

Hint: As a preliminary step, do some angle-chasing. We can even do it mindlessly, without plan. Just fill in all the angles. The only fact we will need is that the angles of a triangle add up to $180^\circ$.

As a start, note that $\angle RQS=45^\circ$, so $\angle VQS=\angle VQR=22.5^\circ$. But $\angle QSV=45^\circ$, so $\angle SVQ=180^\circ-(22.5^\circ+45^\circ)=112.5^\circ$. You may find things less messy if you let $22.5^\circ=a$. Then the angles of $\triangle QSV$ are $a$, $2a$, and $5a$.

Keep filling in angles. When you are finished, or earlier, look for similar triangles. You will notice that $\triangle QSV$ has the same angles as $\triangle QRT$. So these two triangles are similar.

Your desired equality is a direct consequence of this similarity. It comes more naturally as $\frac{SV}{QV}=\frac{TR}{QR}$. Or perhaps note that $SV$ and $TR$ are corresponding sides of our two similar triangles, as are $QV$ and $QT$, so $\frac{SV}{TR}=\frac{QV}{QT}$.