Pedal Triangle

The pedal triangle of a triangle ABC is a triangle formed by dropping perpendiculars from the vertices of triangle ABC to the opposite sides.

Area of Pedal Triangle

The area of the pedal triangle of a triangle ABC with circumradius R is given by:

(1/2) R2sin 2A sin 2B sin 2C

where A, B, and C are the angles of triangle ABC.

Explanation

Let P, Q, and R be the feet of the perpendiculars from vertices A, B, and C of triangle ABC to the opposite sides, respectively. Let O be the circumcenter of triangle ABC.

We know that the circumradius R is the distance from the circumcenter O to any vertex of triangle ABC. Therefore, we can write:

AP = BP = CP = R

Now, consider the right triangles AOP, BOQ, and COR. We have:

sin A = OP/R, sin B = OQ/R, and sin C = OR/R

Multiplying these equations, we get:

sin A sin B sin C = (OP/R) (OQ/R) (OR/R) = (OP OQ OR)/(R3)

But OP OQ OR is the volume of the parallelepiped formed by vectors OA, OB, and OC, which is twice the area of triangle ABC.

Therefore, we can write:

sin A sin B sin C = 2 [ABC]/R3

where [ABC] is the area of triangle ABC.

Now, let D, E, and F be the midpoints of sides BC, CA, and AB, respectively. We know that the pedal triangle of triangle ABC is similar to triangle DEF, with ratio of similitude equal to the circumradius R.

Therefore, the area of the pedal triangle is (1/2) R2 times the area of triangle DEF. But triangle DEF is the medial triangle of triangle ABC, so its area is (1/4) [ABC].

Therefore, we have:

Area of Pedal Triangle = (1/2) R2 [DEF] = (1/2) R2 (1/4) [ABC] = (1/8) R2 [ABC]

Substituting the expression for sin A sin B sin C from above, we get:

Area of Pedal Triangle = (1/16) R2 sin 2A sin 2B sin 2C

But we know that sin 2A = 2 sin A cos A, sin 2B = 2 sin B cos B, and sin 2C = 2 sin C cos C. Substituting these expressions, we get:

Area of Pedal Triangle = (1/2) R2 sin A sin B sin C cos A cos B cos C

Finally, using the identity cos A cos B cos C = (s2 - r2 - 4Rr)/(4R2), where s is the semiperimeter of triangle ABC, r is the inradius, and R is the circumradius, we get:

Area of Pedal Triangle = (1/2) R2 sin 2A sin 2B sin 2C