Triangle with sides tangent to a circle

Text

Three lines are tangent to a circle, at the points $A$, $B$ and $C$ respectively, forming the triangle $P\hat{Q}R$. It is known that $PC=2QC$, that the perimeter of triangle $PQR$ is $24$ cm and that $PQ+2QR+3PR=50$ cm. Determine the sides of triangle $PQR$. (figure in the original test: $R$ is the vertex with the two tangents $RA$ and $RB$; from $Q$ (on $RA$) the tangent $QC$; $P$ is the intersection of $QC$ with $RB$.)

Solution

Use the tangent segments (from the same external point the two tangents have equal length). Set: $RA=RB=r,\qquad QA=QC=q,\qquad PB=PC=p.$ The condition $PC=2QC$ gives $p=2q$. Express the sides: $PQ=p+q=3q,\qquad QR=r-q,\qquad RP=r-2q.$ Perimeter: $3q+(r-q)+(r-2q)=2r=24\ \Rightarrow\ r=12$. Second condition: $3q+2(r-q)+3(r-2q)=5r-5q=50\ \Rightarrow\ 60-5q=50\ \Rightarrow\ q=2$ (so $p=4$). Hence $PQ=6,\qquad QR=10,\qquad RP=8.$ Check: perimeter $24$ ✓; $\;6+20+24=50$ ✓; $\;PC=4=2\,QC$ ✓. $\boxed{PQ=6,\quad RP=8,\quad QR=10.}$