Finding Triangle Vertices from MidpointsDate: 09/18/1999 at 15:16:10 From: Wing-Shan Subject: Midpoint coordinates Hi, If you know the coordinates of the midpoints of the sides of a triangle, how can you find the coordinates of its vertices? Date: 09/18/1999 at 15:50:22 From: Doctor Floor Subject: Re: Midpoint coordinates Hi, Wing-Shan, Thanks for your question. Suppose the midpoints of the sides of a triangle are given by (a1,a2), (b1,b2) and (c1,c2). The centroid of the triangle formed by these midpoints is given by: ([a1+b1+c1]/3,[a2+b2+c2]/3) This is also the centroid of the original triangle (can you see why?). The centroid of a triangle divides the medians of that triangle in the ratio 1:2, where the side midpoints are closer to the centroid than the vertices. So if we make a step from (a1,a2) to ([a1+b1+c1]/3,[a2+b2+c2]/3), and then let that step be followed by two steps of the same size, we find one of the vertices of the original triangle. The first step from (a1,a2) to ([a1+b1+c1]/3,[a2+b2+c2]/3) is: [(-2a1+b1+c1)/3] [(-2a2+b2+c2)/3] (this means, add (-2a1+b1+c1)/3 to the x-coordinate, and (-2a2+b2+c2)/3 to the y-coordinate). The total number of steps to the vertex of the original triangle is three times as much: [ -2a1+b1+c1 ] [ -2a2+b2+c2 ] We start from (a1,a2). So the vertex of the original triangle is found as: (-a1+b1+c1,-a2+b2+c2) And in the same way the other two vertices are found as: (a1-b1+c1,a2-b2+c2) and (a1+b1-c1,a2+b2-c2). I hope this helped. If have more questions, just write us back. Best regards. - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ |
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