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December 7, 2019

Coordinates of special points of the 3-4-5 triangle

Filed under: mathematics — ckrao @ 3:40 am

One thing I observed is that the 3-4-5 triangle is rather attractive in solving problems using coordinates. If the vertices are placed at (0,3), (0,0) and (4,0) the following are the coordinates of points and equations of some lines of interest.

Line AC: x/4 + y/3 = 1

Incentre: (1, 1)

Centroid: (4/3, 1)

Circumcentre: (2, 1.5)

Orthocentre: (0, 0)

Nine-point centre: (1, 3/4) (midpoint of the midpoints of AB and BC)

Angle bisectors: y = x, y=-2x +3, y=4/3-x/3

Ex-centres (intersection of internal and external bisectors): (3,- 3), (6, 6), (-2, 2)

3-4-5

Lines joining the excentres (in red above): y=-x, y=x/2 +3, y = 3(x-4)

Altitude to the hypotenuse: y = 4x/3

Euler line: y=3x/4

Foot of altitude to the hypotenuse: (36/25, 48/25) (where x/4 + y/3 = 1 intersects y=4x/3)

Symmedian point (midpoint of the altitude to the hypotenuse [1]): (18/25, 24/25)

Contact points of incircle and triangle: (1,0), (0,1), (8/5, 9/5)

Gergonne point (intersection of Cevians that pass through the contact points of the incircle and triangle = the intersection of y=3-3x and y=1-x/4): (8/11, 9/11)

Nagel point (intersection of Cevians that pass through the contact points of the ex-circles and triangle = the intersection of y=3-x and y=2-x/2: (2,1)

Reference

[1] Weisstein, Eric W. “Symmedian Point.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/SymmedianPoint.html

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