let AB and CD be perpendicular diameters of a circle. two chords AE and AF, which lie in the two different semicircles determined by the diameter AB are cut by the diameter CD in the points X and Y respectively. prove that angle EXF = angle EYF

To prove that angle EXF is equal to angle EYF, we need to use some properties of diameters and angles formed by chords in a circle.

1. First, let's draw the diagram described in the problem:
- Draw the circle with diameter AB.
- Draw the perpendicular diameter CD that intersects AB at its center.
- Draw two chords AE and AF that lie in the two different semicircles determined by AB.
- Let X be the point where AE intersects CD and Y be the point where AF intersects CD.
- We need to prove that angle EXF is equal to angle EYF.

2. Since AB and CD are perpendicular diameters of the circle, AC is a diameter as well. Therefore, AE is a chord in the semicircle determined by diameter AC, and AF is a chord in the semicircle determined by diameter BC.

3. In a circle, if a chord (AE) and a diameter (CD) intersect at a point (X), then the angle between the chord and the diameter at that point (angle EXC) is a right angle (90 degrees).

4. Similarly, in a circle, if another chord (AF) and the same diameter (CD) intersect at a point (Y), then the angle between the chord and the diameter at that point (angle FYC) is also a right angle (90 degrees).

5. Now, consider the quadrilateral AEFC:
- Angle AXE is an exterior angle at point X for triangle AXC.
- Therefore, angle AXE is equal to the sum of the two remote interior angles, which are angles EXC and AXC.
- Angle AXE = angle EXC + angle AXC = 90 degrees + angle AXC (from step 3).

6. Similarly, consider the quadrilateral AEFY:
- Angle FYA is an exterior angle at point Y for triangle FYC.
- Therefore, angle FYA is equal to the sum of the two remote interior angles, which are angles FYC and AYC.
- Angle FYA = angle FYC + angle AYC = 90 degrees + angle AYC (from step 4).

7. Since angle AXC and angle AYC are opposite angles created by intersecting lines, they are equal. Therefore, we can write angle AXE = 90 degrees + angle AXC = 90 degrees + angle AYC = angle FYA.

8. Now, let's consider the angles EXF and EYF:
- Angle EXF can be expressed as the sum of two angles: angles AXE and angle AXF.
- Angle EYF can be expressed as the sum of two angles: angles FYA and angle FYE.

9. From step 7, we know that angle AXE = angle FYA. So we can replace these angles in the expressions for EXF and EYF:
- EXF = (angle AXE) + angle AXF = (angle FYA) + angle AXF (from step 7).
- EYF = (angle FYA) + angle FYE.

10. Since both angle AXE and angle FYA are equal, we can conclude that:
- EXF = EYF (from step 9).

Therefore, we have proved that angle EXF is equal to angle EYF.