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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).

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Q: Can bisector of angle divide the triangle into similar triangle?
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