Use compass and ruler to draw on paper the construction described in the app below.
The following app is the same as the previous one, but now includes GeoGebra tools.[br][br]
Explore the entire construction in the app above, then use the GeoGebra tools to measure the sides of the triangle and verify the construction numerically.[br][br](Use the [i]Undo [/i]and [i]Redo [/i]buttons at the top right of the toolbar, or refresh the browser page to delete possible objects you have created but that are not useful or correct).
Describe the properties of isosceles triangles.
Referring to the construction above, explain why the triangle you obtain is isosceles.
Let [math]\bigtriangleup ABC[/math] be an isosceles triangle with vertex [math]A[/math].[br]On the extension of [math]AC[/math], beyond [math]A[/math], choose a point [math]D[/math], and on the extension of [math]AB[/math], beyond [math]A[/math], choose a point [math]E[/math] such that [math]AD\cong AE[/math].[br][br](a) Prove that [math]DB\cong CE[/math][br](b) The extensions of [math]DB[/math] and [math]CE[/math] intersect at a point [math]P[/math]. Prove that [math]\bigtriangleup PBC[/math] is isosceles.[br](c) Draw the ray [math]PA[/math]. Prove that [math]\bigtriangleup ADP\cong\bigtriangleup AEP[/math].[br](d) Prove that the ray [math]PA[/math] is the bisector of [math]\angle CAB[/math].
If a statement is false, correct it to make it true, or provide a counterexample.[br][br][list=1][*]The base angles of an isosceles triangle can be obtuse.[/*][*]A right triangle cannot be isosceles.[/*][*]There is no such thing as an obtuse isosceles triangle.[/*][*]In an isosceles triangle, the altitude relative to the oblique side is also a median.[br][/*][/list]