Anti-Thales In-Class

Class 11/2

Quiz 1 - 20 minutes

Similar triangles that share an angle - Thales and anti-Thales

We are very familiar with one situation where two triangles share the same angle; this is the Thales figure. If OAB is a triangle and A' is on ray OA and B' is on ray OB, with OA'/OA = OB'/OB, then triangle OA'B' is similar to triangle OAB, and the lines AB and A'B' are parallel.

But it is also possible to have similar triangles when A' is on ray OA and B' is on ray OB, but instead of triangle OA'B' being similar to triangle OAB, it is similar to OBA but not OAB. In this case line A'B' is not parallel to line AB. We call this case an anti-Thales figure.

To Prove the Triangles are Similar and form an Anti-Thales Figure:
The angle at vertex O is the same for each of the two triangles.

To apply the AA criterion it suffices to find another pair of equal angles, for example show angle OAB = angle OBA.

To apply the SAS criterion, it is enough to show that OA'/OB = OB'/OA since the included angles at O are equal.

Note 1. It may be that either B = B' or A = A' in this figure.
Note 2. A more general case which we will also include are when A' is on line OA but O is between A and A' and also B' is on line OB but O is between B and B' (careful: either both A' and B' are on opposite sides of O from A and B or both are on the same side, not a mix).
Note 3. The "anti-Thales" terminology is peculiar to this course. Sometimes this relationship is called anti-parallel, but that leads to confusion, since it is not much like parallelism at all.

Example 1. Altitude in a right triangle.

Let ABC be a right triangle with C the right angle. Let D be the foot of the altitude from C to the hypotenuse AB. Then the similar triangles ABC and ACD form an anti-Thales figure with common angle A. [The triangles are similar by AA because angle ACB = angle ADC are right angles.]

Also there is a second anti-Thales figure with common angle B, giving triangle ABC similar to triangle CBD. Both were used in the proof of the Pythagorean Theorem.

Example 2. Triangles formed by intersecting chords.

In a circle c, let AB and CD be chords intersecting at point P inside c. Then the two triangles ACP and DBP have equal angles at P (vertical angles). Show that this is an anti-Thales situation by using the inscribed angle theorem to find other equal angles. [Angle CAB = angle CDB because they are inscribed angles with the same included arc. Since angle CAP = angle CAB = angle CDB = angle PDB.] What relation among PA, PB, PC, PD does this give?

Example 3. Triangles formed by intersecting secants.

In a circle c, let AB and CD be secant lines that intersect at a point P outside of c.Then again show that the two secants form an anti-Thales figure with two similar triangles. What relation among PA, PB, PC, PD does this give?

Follow this link to a dynamic Java Sketchpad web page showing Ex. 2 and Ex. 3.

(These examples are problems 35 and 36, page 151 of B&B.)