100 followers question

Geometry Level 5

Let Δ P Q R \Delta PQR be the pedal triangle of Δ A B C \Delta ABC . Excenters of Δ P Q R \Delta PQR are ( 20 , 8 ) , ( 4 , 12 ) , ( 13 , 1 ) (20,8), (4,12), (13,1) .

If one of the vertices of Δ P Q R \Delta PQR is ( 14 , 2 ) , (14,2), then find the area of Δ A B C \Delta ABC .

Details and Assumptions:

  • Pedal triangle of a triangle is formed by joining feet of altitudes to the sides of the triangle.
  • An excenter of a triangle is a point of intersection of an internal angle bisector and two external angle bisectors of the triangle.

Note: Try to solve this within a minute.


The answer is 70.

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1 solution

Excentres of pedal triangle of a triangle are nothing but the vertices of triangle itself. Hence area of Δ A B C = 1 2 × 20 8 1 4 12 1 13 1 1 = 70 s q . u n i t s \Delta ABC = \dfrac { 1 }{ 2 } \times \begin{vmatrix} 20 & 8 & 1 \\ 4 & 12 & 1 \\ 13 & 1 & 1 \end{vmatrix} = 70 sq. units

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