The theorem that led to the proof of the Nine Point Circle was first printed in Thomas Leybourn's Mathematical Repository, in 1804, on page 18. It is printed as follows:
VII. QUESTION 67, by Mr. BENJAMIN BEVAN.
In a plane triangle, let w be the centre of a circle passing through x, y, and z; then will Cw = Cc, and be in the same right line; and wx = wy = wz = 2R, or the diameter of the circumscribing circle; where x, y, z, &c. represent the same points and lines as they denote in the Synopsis of Data for the Construction of Triangles.
The proof of the theorem was given by John Butterworth, on page 143 of the same volume. The proof provided the basis of the proof of the Nine Point Circle. It is as follows:
VII. QUESTION 67, by Mr. BENJAMIN BEVAN.
In a plane triangle, let w be the centre of a circle passing through x, y, and z; then will Cw = Cc, and be in the same right line; and wx = wy = wz = 2R, or the diameter of the circumscribing circle; where x, y, z, &c. represent the same points and lines as they denote in the Synopsis of Data, for the Construction of Triangles?
SOLUTION, by Mr. JOHN BUTTERWORTH, Haggate.
Let S, H, and G (fig. 114. pl. 6.) be the points where the lines xc, yc, and zc meet the circumscribing circle, and draw the radii CS, CH, and CG; also draw xw parallel to CS meeting cC produced in w, and join yw, sw; then w is the centre of a circle passing through x, y, and z. For it is now well known that Sc = Sx, therefore Cc = Cw and CH = Hy, consequently yw is parallel to CH. But CH is = CS, therefore yw is = wx = 2CS. In like manner it is proved that zq = xw = 2CS; therefore w is the centre of a circle passing through the points x, y, and z, and consequently the points c, C, w, are in a straight line, and Cc = Cw; and yw = xw = 2CS = 2R. Q.E.D.
Thus nearly was the proposition demonstrated by Messrs. Boole, Dawes, and Johnson.