** **

**11. Construct any acute triangle ABC and its circumcircle. **

**Construct the three altitudes AD, BE, and CF. Extend each
altitude**

** to its
intersection with the circumcircle at corresponding points **

**P, Q, and R.**

** **

Find .___(1)

We can know the result is 4 by calculating GSP.

If a triangle is a regular triangle, we can easily know the result

of is 4. Because in a regular triangle the
circumcenter

is congruent to the orthocenter and a distance from a vertex to a

circumcenter is two times of a distance from a circumcenter to

an opposite side.

What is the value of in acute triangle ABC?

LetÕs prove this result.

We can find the result from the pedal triangle DEF and a triangle
PQR.

Length of
HD=Length of DP

Because a triangle
DEF is similar to a triangle PQR and the ratio is 1:2.

So we change the equation (1) the following.

We can get the following result about the area of a triangle ABC.

**Area of ABC**= BC * AD * 1/2 ---(1)

= AC * BE * 1/2 ----(2)

= AB * CF * 1/2 ----(3)

And **area of ABC** = area of HBC + area of HCA + area
of HAB

= BC*HD*1/2+AC*HE*1/2+AB*HF*1/2
Ð (4)

BC * AD = BC * HD+AC * HE+AB * HF

—

By (1),(2) and (3)

—

—

So we can know the
result of equation (1)

= 4

We can apply this result to the obuse triangle.

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