### On an instrument for trisecting any angle

Jas. N. Miller
1901 Proceedings of the Edinburgh Mathematical Society
FIGURE 5. 2. Antiparallel to sides, (a) Let O and O' be two points, and let the intercepts cut off be by lines drawn antiparallel to the sides. AODE is isosceles. .-. ~ = cotODE = cot A. 2x .: a = 2a;cotA, (3 = 2ycotB, etc. a' = 2a;'cot A, etc. .-. ax' = 2xx'cotA = a'x or AO'DE = AOD'E', / V = 2y2/'cotB=£'// or AO'FG = AOFG', yz = 2«s'cotO = y'z or AO'HI * AOH'I'; and the six triangles will be equal if the conditions a;a:'cotA = yy'cotH = zz'cotC are fulfilled, or if xx':yy :zz : : tanA: tanB :
more » ... tanC. In analogy to parallels and antiparallels such a pair of points might be called antireciprocal points. Now if O be the orthocentre x:y:z : : : -=r : -pr, cosA cosB cosC and if 0' be the symmedian point x :y :z : : sinA : sinB : sinC. Hence for these two points the six triangles are equal. (6) The intercepts made on the sides by lines drawn antiparallel to the sides will be equal, if the point through which they are drawn has its perpendiculars on the sides in the ratio tanA : tanB : tanC. If a = /3 = y, then axotA = ycotB = zcotC, 1 1 1 •"' X : y : Z~c o t A : c o t B : c o t C = tanA : tanB : tanC. Hence such a point might be called antireciprocal point to the incentre.