SMH Geometry: Soddy Circles Multi-Cache

When three circles are mutually tangential, there are two other circles that are tangential to all three. These are known as the inner and outer Soddy circles. A brief history of these can be found here and a full history is available here. Visit this page for an animation of these circles.

In 2001, David Eppstein published a straightedge and compass method for finding the centers of these circles. The following is a paraphrase of that method:

1. Draw the triangle joining the centers of the circles.
2. For each vertex, draw the diameter of its circle that is perpendicular to the opposite side. Mark the endpoints of the diameter.
3. Draw a line through each endpoint to the point of tangency of the other two circles. Mark the points where each line intersects its circle a second time. These are the points of tangency to the inner and outer Soddy circles.
   • The three points inside the triangle lie on the circumference of the inner Soddy circle.
   • The three points outside the triangle lie on the circumference of the outer Soddy circle.

1. At the posted coordinates, you will find three pairs of coordinates. Use the ones labelled “WP2=A” and “WP3=B”. Ignore the one labelled “C”.
2. At WP2, you will find the North coordinate of each Soddy center.
3. At WP3, you will find the West coordinate of each Soddy center.
4. To find the final tag, do the following:
   • Determine the distance and bearing from the inner Soddy center to the outer Soddy center.
   • Round the bearing to the nearest even degree.
   • Multiply the distance by 5.17.
   • Project that distance and bearing from the inner Soddy center. Set the last digit of the west coordinate to 1.
5. The final tag has the coordinates of the cache. The tag is on a large pine tree within 5 m of the trail. The cache is within 100m of the tag.

This is a very calculation dense puzzle. It’s OK to read the tags and go home and solve the puzzle using Mapsource and Fizzycalc or similar software.

At the posted coordinates, there are three pairs of coordinates. They contain the coordinates of the three vertices of a triangle. Use the method described below to find the Soddy centers and then the cache. It is David Eppstein’s method adapted for the GPS.

Part 1: Given the vertices of a triangle, A, B and C, find the points of tangency of the circles centered on those vertices.

1. Find the length and bearing of each side in meters: AB, AC and BC.
2. Calculate the radius of the circle centered on A: rA=((AB+AC-BC))/2.
3. Calculate the radius of the circle centered on B: rB = AB-rA.
4. Calculate the radius of the circle centered on C: rC = AC-rA.
5. From B, project a waypoint rB meters towards C. Call it D.
6. From A, project a waypoint rA meters towards C. Call it E.
7. From A, project a waypoint rA meters towards B. Call it F.
8. Optional: To see the three tangential circles:
   • For each vertex, create a proximity circle using its radius.

9. From the center of each circle, using its radius, project two waypoints perpendicular to the opposite side. One goes away from the opposite side, the other goes toward it. Call the outer points G, H and I, and the inner points J, K and L. Make a note of the bearing from each point to its center (i.e. exactly opposite to the direction of the projection) and call them bGA, bHB, etc.

10. For the circle centered on A, do the following:
    • Find the bearing from G to the point of tangency of the other pair of circles, D. Call it bGD.
    • Calculate a new bearing, bAM, where bAM = 2×bGD – bGA.
    • From the center, A, project a waypoint using its radius, rA, at the bearing bAM. Call the point M. M is one vertex of the inner triangle.
11. Repeat step 10 with B, H and E to get N, and with C, I and F to get O.
12. The center of the inner Soddy circle is at the circumcenter of the triangle formed by M, N and O. Refer to GC3K4NM for instructions on how to calculate the circumcenter. Call it X.
13. Optional: To see the inner Soddy circle:
    • Calculate the radius by determining the distance from the center, X, to one of the vertices M, N or O. Call it rX.
    • Create a proximity circle centered on X using rX.

14. For the circle centered on A, do the following:
    • Find the bearing from the point of tangency of the other pair of circles, D, to J. Call it bDJ.
    • Calculate a new bearing, bAP, where bAP = 2×bDJ – bJA.
    • From the center, A, project a waypoint using its radius, rA, at the bearing bAP. Call the point P. P is one vertex of the outer triangle.
15. Repeat step 14 with B, K and E to get Q, and with C, L and F to get R.
16. The center of the outer Soddy circle is at the circumcenter of the triangle formed by P, Q and R. Refer to GC3K4NM for instructions on how to calculate the circumcenter. Call it Y.
17. Optional: To see the outer Soddy circle:
    • Calculate the radius by determining the distance from the center, Y, to one of the vertices P, Q or R. Call it rY.
    • Create a proximity circle centered on Y using rY.

18. Use steps #4 and #5 of the multi to find the final tag and the cache location.

• Optional checks:
  • The distance from each vertex of the inner triangle to its circumcenter should be:
    (rA∙rB∙rC)/(rA∙rB+rB∙rC+rC∙rA+2√(rA∙rB∙rC(rA+rB+rC)))
  • The distance each vertex of the outer triangle to its circumcenter should be:
    -(rA∙rB∙rC)/(rA∙rB+rB∙rC+rC∙rA-2√(rA∙rB∙rC(rA+rB+rC)))
• If proximity circles work well on your GPS, then an alternate method for step 9 is create a route joining the diameter’s outer endpoint to the point of tangency of the other two circles and find where it intersects the proximity circle. Step 13 will be similar, except that you’ll need to project a waypoint through the diameter’s inner endpoint and then route to that point in order to see where it intersects the circle. If you use this method, please say so in your log and indicate what model of GPS you used.
• If you have a more efficient method for doing this on a GPS, I would appreciate knowing about it.
• If you’d like to practice this, try it out on the following: if the vertices of the triangle are at N45 18.599 W75 47.178, N45 18.220 W75 47.308, and N45 18.405 W75 46.715, the last tag is at N 45 17.867 W 75 46.815.

Except for the final approach to the cache, the route is entirely on trails. All tags are on the backs of trees beside the trail. The cache is off trail in an easily accessed area within 100m of the final tag. The terrain rating reflects the fact that the trails are occasionally rocky and/or steep. Just watch your footing.

The trails are excellent for hiking, mountainbiking, skiing and snowshoeing.