Yes.
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Given certain triangles, it would be possible for an angle to be bisected and create two new triangles which are similar to each other. And in the case of a [45°, 45°, 90°] right triangle, if you bisect the right angle, then you will create two new [45°, 45°, 90°] triangles (both similar to each other and similar to the original).
In geometry when comparing two triangles, if all three angles of each triangle are congruent to corresponding angles in the other triangle, then both triangles are similar.
If the 2 triangles are right triangles, which are congruent to slicing the rectangle on the diagonal, then arrange one on top of the rectangle, and the other to the side, so that the two hypotenuses are in line with each other. This will make a bigger right triangle, which is similar to the smaller right triangles - each side is double of the smaller triangles.
No, not all isoceles triangles can be equilateral triangles because an equilateral triangle has sides that are all equal to each other and an isoceles triangle has only two sides that are equal to each other.
This is known as the Sierpinski triangle.