## Some Relationships in a Right Triangle

On right triangle ABC with right angle at C, construct squares
CBDE and ACFG.

Construct AD and BG intersecting at H.

Draw altitude CK through H to AB.

Click **here** for a GSP Sketch of
this configuration.

**Prove**: (1)

**Help**

**Prove**: (2)

and
which is equivalent to

**Help**

That is, PC and QC are half the **harmonic
mean** of the legs of right triangle ABC.

**Prove**: (3)

The altitude CK is **concurrent** with AD and BG at H.

Hint: Use **Ceva's
Theorem**.

**Help**

**Prove**: (4)

Hint: Use **Menelaus's
Theorem**

Note: This means CH is one half the Harmonic mean of the altitude
and the hypothenuse of the triangle ABC.

**Help**

.

**Prove**: (5)

CH is the length of the side of a square inscribed in triangle
ABC with one side lying along AB.

Help **See Square
Inscribed along a Base of Any Triangle**

For any triangle the length **s** of the inscribed square
along a given base is given by the formula

where **a** is the length of the base and **h** is the
altitude to the the base. Thus in the current problem CH = **s**,
AB = **a**, and CK = **h**.

Special thanks to Jim Metz of Honolulu for calling
these problems to my attention, discussing solutions, and raising
questions.

Return to the **EMAT
4600/6600 Page**

Reference:

Charosh, M. (1965) *Mathematical Challenges *Washington,
DC: National Counctil of Teachers of Mathematics.