# 4D Geo

That’s odd, these problems are so hard! Where are they even from?

Geo????? → Geo?????? → Geo???? → Geo?????

In what follows, triangle PQR is an acute triangle, and point S is in the interior of triangle PQR.

1. Given that Q = (171.316,−92.366), R = (6.258,−264.479), S = (48.501,−139.162), and that S is the centroid of PQR, find P.
2. Given that P = (−243.537,179.039), Q = (338.676,341.390), R = (−30.976,−36.228), and that S is the incenter of PQR, find S.
3. Given that P = (27.908,−102.106), Q = (439.054,296.796), R = (−73.114,203.114), and that S is the orthocenter of PQR, find S.
4. Given that P = (−243.537,179.039), R = (−30.976,−36.228), S = (−56.792,67.804), and that S is the incenter of PQR, find Q.
5. Given that P = (27.908,−102.106), Q = (439.054,296.796), S = (18.153,−91.237), and that S is the orthocenter of PQR, find R.
6. Given that P = (79.667,101.833), Q = (−20.136,−72.144), R = (−17.585,114.724), and that S is the symmedian point of PQR, find S.
7. Given that P = (79.667,101.833), R = (−17.585,114.724), S = (33.942,89.051), and that S is the symmedian point of PQR, find Q.
8. Given that Q = (439.054,296.796), R = (−73.114,203.114), S = (3.950,122.419), and that S is the orthocenter of PQR, find P.
9. Given that P = (−243.537,179.039), Q = (338.676,341.390), S = (20.520,106.335), and that S is the incenter of PQR, find R.
10. Given that Q = (−20.136,−72.144), R = (−17.585,114.724), S = (9.098,−61.302), and that S is the symmedian point of PQR, find P.
11. Given that P = (15.056,−80.644), R = (6.258,−264.479), S = (22.241,−142.971), and that S is the centroid of PQR, find Q.
12. Given that Q = (338.676,341.390), R = (−30.976,−36.228), S = (35.932,−30.136), and that S is the incenter of PQR, find P.
13. Given that P = (79.667,101.833), Q = (−20.136,−72.144), S = (19.049,−70.431), and that S is the symmedian point of PQR, find R.
14. Given that P = (27.908,−102.106), R = (−73.114,203.114), S = (34.156,−99.532), and that S is the orthocenter of PQR, find Q.
15. Given that P = (15.056,−80.644), Q = (171.316,−92.366), R = (6.258,−264.479), and that S is the centroid of PQR, find S.
16. Given that P = (15.056,−80.644), Q = (171.316,−92.366), S = (84.647,−46.540), and that S is the centroid of PQR, find R.

 m3 − m10 m8 + m12 m2 m16 Km15 − m11 Im12 − m11 Hm9 − m2 Gm13 + m16 m6 − m7 m4 − m11 m5 + m7 m3 − m15 m1 − m12 m12 m14 − m2