Given the midpoints of a pentagon, construct the pentagon

已知中心五边形，作五边形 苏剑林 · 2010-07-25 · Mathematics polygon compass and straightedge construction

This is another problem from the Shulian Tiandi Forum, a math forum that's really well put together. ^_^

Given five fixed points A, B, C, D, E, construct pentagon FGHIJ such that the midpoints of its five sides are exactly these five given points.

Pentagon problem more

As usual, BoJone tackles this kind of problem using coordinates. As shown in the figure below, we place the five given vertices in a coordinate system, with side DE as the x-axis and E as the origin, and let the five points be A(a,b); B(c,d); C(e,f); D(g,0); E(0,0).

Pentagon problem – coordinate diagram

If we can determine the coordinates of any one vertex of pentagon FGHIJ, then the rest of the pentagon is easy to construct. So let's try to find point J. Let J have coordinates (x,y). Since J, A, F are collinear and JA=AF, we get F's coordinates as (2a-x,2b-y); similarly, G's coordinates are (2c-2a+x,2d-2b+y); H's coordinates are (2e-2c+2a-x,2f-2d+2b-y); I's coordinates are (2g-2e+2c-2a+x,-2f+2d-2b+y); and J's coordinates are (-2g+2e-2c+2a-x,2f-2d+2b-y).

But remember, we originally set J=(x,y), so we must have:

-2g+2e-2c+2a-x=x

2f-2d+2b-y=y

Solving these gives x=-g+e-c+a, y=f-d+b. My goodness! Constructing this is really that simple! All you need to do is add and subtract a few known quantities.

Here's a brief description of the construction:

1. Take ED as the x-axis, E as the origin, and draw the y-axis.
2. On the y-axis, plot the projections of AE, BE, and CE separately.
3. On the y-axis, construct the segment equal to "
the sum of the projections of AE and CE
minus the projection of BE."
4. On the x-axis, plot the projections of AE, BE, and CE separately.
5. On the x-axis, construct the segment equal to "
the sum of the projections of AE and CE
minus
the sum of the projection of BE and ED
".
6. Using the coordinates fixed by steps 3 and 5, construct point J.
6. Draw line JA, and mark off AF=JA on it; then proceed similarly for the remaining steps... until the whole pentagon is drawn!