Help with a geometry problem

Pythagorean Theorem · #1 $Old$ November 1st, 2009, 11:36 AM

Given:
BAL is a right triangle at A.
O is the midpoint of [BL].
The circle of diameter [BO] cuts [BA] at I.

Show that I is the midpoint of [AB].

earboth · #2 $Old$ November 1st, 2009, 12:44 PM

Quote:

Originally Posted by Pythagorean Theorem $View Post$

Given:
BAL is a right triangle at A.
O is the midpoint of [BL].
The circle of diameter [BO] cuts [BA] at I.

Show that I is the midpoint of [AB].

1. Triangle BIO is a right triangle with $\angle(BIO) = 90^\circ$ because I is placed on the Thales circle over BO.

2. You have 2 similar triangles: $\Delta(BAL) \sim \Delta(BIO)$

3. Use proportions.

Plato · #3 $Old$ November 1st, 2009, 01:25 PM

Here is a second way.
The $\angle OIB$ being inscribed in a semi-circle is a right angle.
Therefore, $\overleftrightarrow {AL}||\overleftrightarrow {BI}$ so by the midpoint theorem $I$ is the midpoint of $\overline {AB}$ .