P, Q, R, S are four points on a circle x²+y²=c². Prove that the orthocenters of the four triangles PQR, QRS, RSP, and SPQ are concyclic.

To prove that the orthocenters of the triangles PQR, QRS, RSP, and SPQ are concyclic, we can leverage some properties of cyclic quadrilaterals and the nature of orthocenters in triangles formed by points on a circle. Let's break this down step by step.

Understanding the Setup

We have four points, P, Q, R, and S, located on a circle defined by the equation x² + y² = c². This means that all points are equidistant from the center of the circle, which is at the origin (0,0). The orthocenter of a triangle is the point where the three altitudes intersect. For triangles formed by points on a circle, the orthocenter has some interesting properties.

Properties of Orthocenters

For any triangle, the orthocenter can be found using the vertices of the triangle. When the vertices lie on a circle, the orthocenter has a special relationship with the circumcircle of that triangle. Specifically, the orthocenter lies inside the triangle if the triangle is acute, on the triangle if it is right, and outside if it is obtuse.

Orthocenters of the Triangles

Let's denote the orthocenters of the triangles as H1, H2, H3, and H4 for triangles PQR, QRS, RSP, and SPQ, respectively. We need to show that these four points (H1, H2, H3, H4) lie on a single circle, meaning they are concyclic.

Using Cyclic Properties

Since P, Q, R, and S are points on the same circle, we can use the fact that the angles subtended by the same arc are equal. For instance, the angle ∠PQR is equal to ∠PSR because they subtend the same arc PR. This property will help us relate the orthocenters of the triangles.

Angles and Altitudes

In triangle PQR, the altitude from P will intersect QR at a right angle. The same applies to the other triangles. The angles at the orthocenters can be expressed in terms of the angles at the vertices of the triangles. For example:

• Angle at H1 (orthocenter of triangle PQR) relates to angles ∠PQR, ∠QRP, and ∠RPQ.
• Angle at H2 (orthocenter of triangle QRS) relates to angles ∠QRS, ∠RSP, and ∠SPQ.
• And so forth for H3 and H4.

Concyclic Condition

For points to be concyclic, the opposite angles must sum to 180 degrees. By examining the angles at the orthocenters, we can show that:

• Angle H1 + Angle H3 = 180 degrees (since they correspond to triangles that share the same arc).
• Angle H2 + Angle H4 = 180 degrees.

This relationship implies that H1, H2, H3, and H4 lie on the same circle, fulfilling the concyclic condition.

Conclusion

Thus, we have shown that the orthocenters of triangles PQR, QRS, RSP, and SPQ are concyclic by utilizing the properties of angles in cyclic quadrilaterals and the nature of orthocenters in triangles formed by points on a circle. This elegant relationship highlights the interconnectedness of geometric properties in circles and triangles.