A First Exploration of Isosceles Triangles with Sketchpad

Isosceles Triangles and Circles

An isosceles triangle is a triangle with two (or more) equal sides.  The simplest way to produce such a triangle is to draw a circle with center A through B.  Then construct any point C on the circle.  Draw segments to form triangle ABC.  This triangle is isosceles.  Why?

Isosceles Triangles and an Angle bisector

Draw a triangle ABC as above.  Construct the bisector of angle BAC.  Let D be the intersection of the bisector and BC.

• Measure distances BD and CD.  How do they compare?
• Measure angles BDA and CDA.  How do they compare?

Are the triangles BDA and CDA congruent?  If so, what congruence criterion would you use to prove this?  What would be the S's and what would be the A's?

Isosceles Triangles and a Median Line

Draw a triangle ABC as above.  Construct the midpoint M of segment BC.  Construct segment AM.

• Measure angles BMA and CMA.  How do they compare?
• Measure angles BAM and CAM.  How do they compare?

Are the triangles BMA and CMA congruent?  If so, what congruence criterion would you use to prove this?  What would be the S's and what would be the A's?

Isosceles Triangles and an Altitude

Draw a triangle ABC as above.  Construct the line through A perpendicular to segment BC.  Construct the intersection F of the line and segment BC.

• Measure distances BE and CE.  How do they compare?
• Measure angles BAE and CAE.  How do they compare?

Are the triangles BEA and CEA congruent?  If so, what congruence criterion would you use to prove this?  What would be the S's and what would be the A's?

Isosceles Triangles and a Perpendicular Bisector

Draw a triangle ABC as above.  Construct the midpoint M of BC and construct the line through M perpendicular to segment BC.  

• This line is the perpendicular bisector of BC.  Does it appear to pass through A?

How can you prove it?