area of triangle by 3 common tangentsof 2 circles (2011 Euler Teachers' MO I p4)

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area of triangle by 3 common tangentsof 2 circles (2011 Euler Teachers' MO I p4)

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#1 h Jul 12, 2020, 9:49 AM

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Two disjoint circles are arranged so that one of their common internal tangents are perpendicular to one of their common external tangents. Find the area of the triangle formed by these tangents and the third common tangent of given circles if their radii are $r_1$ and $r_2$ .

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#2 h Jul 12, 2020, 7:32 PM • 1 Y

Y by Mango247

Circle, midpoint $O_{1}(0,r_{1})$ and radius $r_{1}$ .
Circle, midpoint $O_{2}(r_{1}+r_{2},r_{2})$ and radius $r_{2}$ .

The internal tangent line $x=r_{1}$ cuts the external tangent line in the point $S(r_{1},r_{2}+r_{2})$ .
This external tangent line cuts the x-axis (the second external tangent line) in the point $T(\frac{r_{1}(r_{1}+r_{2})}{r_{1}-r_{2}},0)$ .

Area of the triangle $=\frac{r_{1}r_{2}(r_{1}+r_{2})}{r_{1}-r_{2}}$ .

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#3 h Jul 13, 2020, 12:15 PM • 1 Y

Y by Mango247

The line $O_{1}O_{2}$ cuts the line $x=r_{1}$ in the point $K(r_{1},\frac{2r_{1}r_{2}}{r_{1}+r_{2}})$ .

The internal tangent line, through $K$ cuts the x-axis (the second external tangent line) in the point $L(r_{2},0)$ .

The area of the (smaller) triangle $=\frac{r_{1}r_{2}(r_{1}-r_{2})}{r_{1}+r_{2}}$ .

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