Why are $\triangle ABC$ and $\triangle CDE$ similar?

Question

Português: Porque $\triangle ABC$ e $\triangle CDE$ são similares na imagem abaixo? Sendo que $M$ e $N$ são pontos médios de seus segmentos. $AD$ e $BE$ são alturas relativas. $$$$ English: Why are $\triangle ABC$ and $\triangle CDE$ similar in the picture below? Here, $ M $ and $ N $ are midpoints of its segments, and $AD$ and $ BE $ are heights.

Português: Se eu conseguisse mostrar que $\angle BAC =\angle EDC$, é possível?$$$$ English: If I could show that $\angle BAC = \angle EDC$ is possible?

Answer 1

Hint:

• Consider a circle such that $AB$ is its diameter.
• Triangles $\triangle ABE$ and $\triangle ABD$ are right.
• What do you know about inscribed angles or about angles of inscribed quadrilateral?

I hope this helps $\ddot\smile$

Answer 2

Angle AED = angle ADB = 90 degrees (given)

ABDE is a cyclic quadrilateral (converse of angles in the same segment)

Angle BAC = angle EDC (ext. angle cyclic quad)

Similarly, angle ABC = angle DEC

angle C = angle C (common)

Triangle ABC ~ triangle DEC (AAA)

Note1. M and N has nothing to do with this question.

Note2. It is “Triangle ABC ~ triangle DEC” but not “Triangle ABC ~ triangle CDE”. The vertices must match correspondingly.