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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Properties of Triangles

Olympiad workbook · IOQM · Properties of Triangles

Let ABC be a triangle with sides 51,52,53 . Let denote the incircle...

Let ABC be a triangle with sides 51,52,5351,52,53. Let Ω\Omega denote the incircle of △ABC\triangle \mathrm{ABC}. Draw tangents to Ω\Omega which are parallel to the sides of ABC . Let r1,r2,r3\mathrm{r}_1, \mathrm{r}_2, \mathrm{r}_3 be the inradii of the three corner triangles so formed. Find the largest integer that does not exceed r1+r2+r3r_1+r_2+r_3.Let ABC be a triangle with sides 51,52,53 . Let denote the incircle of ABC . Draw tangents to which are parallel to the

Correct answer

15

Step-by-step solution

Let ABC be a triangle with sides 51,52,53 . Let denote the incircle of ABC . Draw tangents to which are parallel to the
Let PQ be one of the required tangents is parallel to BC and meets sides AB and AC at P and Q respectively. Further Let PQ=xP Q=x and BC=51B C=51 Now △ABC\triangle \mathrm{ABC} is similar to △APQ\triangle \mathrm{APQ} so xa=r1r\frac{\mathrm{x}}{\mathrm{a}}=\frac{\mathrm{r}_1}{\mathrm{r}} Similarlyy6=r2rHencer1+r2+r3r=xa+yb+zcNowxa=h−2rh where h is altitude \begin{aligned} \text{Similarly} & \frac{y}{6}=\frac{r_2}{r} \\ \text{Hence} & \frac{r_1+r_2+r_3}{r}=\frac{x}{a}+\frac{y}{b}+\frac{z}{c} \\ \text{Now} & \frac{x}{a}=\frac{h-2 r}{h} \text { where } h \text { is altitude } \end{aligned} xa=1−2Δs⋅2Δ,x=1−as∑xa=3−a+b+cs=1 so r1+r2+r3=r Now r=78(27)(26)(25)78=33⋅52⋅2.132⋅3⋅13=32⋅52=15\begin{array}{ll} & \frac{x}{a}=1-\frac{2 \Delta}{s \cdot 2 \Delta}, x=1-\frac{a}{s} \\ & \sum \frac{x}{a}=3-\frac{a+b+c}{s}=1 \\ \text { so } & r_1+r_2+r_3=r \\ \text { Now } & r=\sqrt{\frac{78(27)(26)(25)}{78}} \\ & =\sqrt{\frac{3^3 \cdot 5^2 \cdot 2.13}{2 \cdot 3 \cdot 13}} \\ & =\sqrt{3^2 \cdot 5^2}=15\end{array}
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