In an acute-angled triangle ABC, the heights AD and CE are drawn, H is the intersection point of the heights.

In an acute-angled triangle ABC, the heights AD and CE are drawn, H is the intersection point of the heights. Prove that points A E D lie on the same circle. Prove that points A E D C lie on one circle

Let point O be the midpoint of the segment AC.

Let’s construct a circle centered at point O and radius AO.

Any point on this circle K is the vertex of the right-angled triangle ACK, because the circle is built on the segment AC, which is its diameter.

Therefore, since triangles AEC and ADC are right-angled triangles, points E and D lie on this circle.

Thus, we have shown that the points A, C, E, D lie on the same circle constructed on the segment AC, both on the diameter and with the center in the middle of this segment, which is what we had to prove.