*Austin Lei, Evan Chen, Moor Xu, Nathan Wong*

This puzzle consists of four different types of “geo”, which comprise the four main steps of the puzzle:

**Geometry**problems, as suggested by the title;**Geography**, subtly suggested by “where” in the flavortext;**Geology**, subtly suggested by “hard” in the flavortext;**Geomancy**, subtly suggested by “even” and “odd” in the flavortext and from the chart given at the end of the puzzle.

We present these four steps in order below.

The first step is to solve all the triangle geometry problems, which requires some computation in Cartesian coordinates.

We comment briefly on the mathematical problem of recovering the third vertex of a triangle given two vertices and the center. In what follows, triangle *ABC* has *A* and *B* given. Then:

- Given the centroid
*G*, the vertex*G*is recovered by*C*= 3*G*− (*A*+*B*). - Given the orthocenter
*H*, the vertex*C*is exactly the orthocenter of triangle*HAB*. - Given the incenter
*I*, the vertex*C*is recovered by reflecting the line*AB*across lines*AI*and*BI*and taking the intersection. - The hardest sub-problem is to find the vertex
*C*given the symmedian point*K*. In fact, such a vertex*C*is not unique in general — there can be up to two — which is why the problem specifies that the triangles are all acute. Fortunately, the problem can be found on Google. See*Constructing the Triangle from Two Vertices and the Symmedian Point*by Michel Bataille, in Forum Geometricorum Volume 18 (2018) 129-133. The completed table is shown here.

# | P.x | P.y | Q.x | Q.y | R.x | R.y | S.x | S.y |
---|---|---|---|---|---|---|---|---|

#1 (G) | -32.071 | -60.641 | 171.316 | -92.366 | 6.258 | -264.479 | 48.501 | -139.162 |

#2 (I) | -243.537 | 179.039 | 338.676 | 341.390 | -30.976 | -36.228 | -31.320 | 122.610 |

#3 (H) | 27.908 | -102.106 | 439.054 | 296.796 | -73.114 | 203.114 | -17.421 | 145.711 |

#4 (I) | -243.537 | 179.039 | 11.549 | 104.939 | -30.976 | -36.228 | -56.792 | 67.804 |

#5 (H) | 27.908 | -102.106 | 439.054 | 296.796 | 12.826 | -85.746 | 18.153 | -91.237 |

#6 (K) | 79.667 | 101.833 | -20.136 | -72.144 | -17.585 | 114.724 | 22.190 | 88.200 |

#7 (K) | 79.667 | 101.833 | 33.673 | -114.227 | -17.585 | 114.724 | 33.942 | 89.051 |

#8 (H) | 41.500 | -82.870 | 439.054 | 296.796 | -73.114 | 203.114 | 3.950 | 122.419 |

#9 (I) | -243.537 | 179.039 | 338.676 | 341.390 | 42.320 | -120.819 | 20.520 | 106.335 |

#10 (K) | 44.091 | -70.301 | -20.136 | -72.144 | -17.585 | 114.724 | 9.098 | -61.302 |

#11 (G) | 15.056 | -80.644 | 45.409 | -83.790 | 6.258 | -264.479 | 22.241 | -142.971 |

#12 (I) | 47.190 | -91.380 | 338.676 | 341.390 | -30.976 | -36.228 | 35.932 | -30.136 |

#13 (K) | 79.667 | 101.833 | -20.136 | -72.144 | 48.400 | -90.350 | 19.049 | -70.431 |

#14 (H) | 27.908 | -102.106 | 49.478 | -94.461 | -73.114 | 203.114 | 34.156 | -99.532 |

#15 (G) | 15.056 | -80.644 | 171.316 | -92.366 | 6.258 | -264.479 | 64.210 | -145.830 |

#16 (G) | 15.056 | -80.644 | 171.316 | -92.366 | 67.569 | 33.390 | 84.647 | -46.540 |

Each of the (X,Y) pairs represents a latitude and longitude pair. We map out which city/lake/farm/etc. is indicated by each pair.

# | Latitude | Longitude | Location name |
---|---|---|---|

#1 | -32.071 | -60.641 | Diamante |

#2 | -31.320 | 122.610 | Talc Lake |

#3 | -17.421 | 145.711 | Topaz |

#4 | 11.549 | 104.939 | (Koh Pich) Diamond Island |

#5 | 12.826 | -85.746 | Apatite |

#6 | 22.190 | 88.200 | Diamond Harbour |

#7 | 33.673 | -114.227 | Quartzsite |

#8 | 41.500 | -82.870 | Gypsum |

#9 | 42.320 | -120.819 | Quartz Mountain |

#10 | 44.091 | -70.301 | Mount Apatite |

#11 | 45.409 | -83.790 | Calcite |

#12 | 47.190 | -91.380 | Corundum Point |

#13 | 48.400 | -90.350 | Topaz Lake |

#14 | 49.478 | -94.461 | Quartz Island |

#15 | 64.210 | -145.830 | Quartz Lake |

#16 | 67.569 | 33.390 | Apatity |

Note that the X-values are simply in increasing order. This suggests we will likely want to re-order these later.

Each of the locations corresponds to the name of a mineral on the Mohs scale of mineral hardness. We record the value of the Mohs entry here.

# | Center | Missing | Mineral | Mohs |
---|---|---|---|---|

#1 | G | P | DIAMOND | 10 |

#2 | I | S | TALC | 1 |

#3 | H | S | TOPAZ | 8 |

#4 | I | Q | DIAMOND | 10 |

#5 | H | R | APATITE | 5 |

#6 | K | S | DIAMOND | 10 |

#7 | K | Q | QUARTZ | 7 |

#8 | H | P | GYPSUM | 2 |

#9 | I | R | QUARTZ | 7 |

#10 | K | P | APATITE | 5 |

#11 | G | Q | CALCITE | 3 |

#12 | I | P | CORUNDUM | 9 |

#13 | K | R | TOPAZ | 8 |

#14 | H | Q | QUARTZ | 7 |

#15 | G | S | QUARTZ | 7 |

#16 | G | R | APATITE | 5 |

At this point we can begin to fill out the shield chart: plugging in the *i*’th hardness into *m _{i}* and evaluate the arithmetic.

Row | Figure position | Calculation | Index |
---|---|---|---|

Row 1 | Fourth daughter | 8 - 5 = | 3 |

Row 1 | Third daughter | 2 + 9 = | 11 |

Row 1 | Second daughter | 1 = | 1 |

Row 1 | First daughter | 5 = | 5 |

Row 1 | Fourth mother (K) | 7 - 3 = | 4 |

Row 1 | Third mother (I) | 9 - 3 = | 6 |

Row 1 | Second mother (H) | 7 - 1 = | 6 |

Row 1 | First mother (G) | 8 + 5 = | 13 |

Row 2 | Fourth niece | 10 - 7 = | 3 |

Row 2 | Third niece | 10 - 3 = | 7 |

Row 2 | Second niece | 5 + 7 = | 12 |

Row 2 | First niece | 8 - 7 = | 1 |

Row 3 | Left witness | 10 - 9 = | 1 |

Row 3 | Right witness | 9 = | 9 |

Row 4 | Judge | 7 - 1 = | 6 |

This gives the following in the shield chart.

3 | 11 | 1 | 5 | 4 | 6 | 6 | 13 |

3 | 7 | 12 | 1 | ||||

1 | 9 | ||||||

6 |

Sort the table of triangle centers and Mohs hardnesses by the triangle center and the omitted point. The letters G, H, I, K are the "usual" letters in Euclidean geometry for the centroid, orthocenter, incenter, and symmedian point, respectively.

# | Center | Missing | Mineral | Mohs | Parity |
---|---|---|---|---|---|

#10 | K | P | APATITE | 5 | odd |

#7 | K | Q | QUARTZ | 7 | odd |

#13 | K | R | TOPAZ | 8 | even |

#6 | K | S | DIAMOND | 10 | even |

#12 | I | P | CORUNDUM | 9 | odd |

#4 | I | Q | DIAMOND | 10 | even |

#9 | I | R | QUARTZ | 7 | odd |

#2 | I | S | TALC | 1 | odd |

#8 | H | P | GYPSUM | 2 | even |

#14 | H | Q | QUARTZ | 7 | odd |

#5 | H | R | APATITE | 5 | odd |

#3 | H | S | TOPAZ | 8 | even |

#1 | G | P | DIAMOND | 10 | even |

#11 | G | Q | CALCITE | 3 | odd |

#16 | G | R | APATITE | 5 | odd |

#15 | G | S | QUARTZ | 7 | odd |

Placed these into the four houses in the upper right of the geomancy shield; following geomancy tradition we only care about the parity of the hardnesses.

For example: the head, neck, body and feet of the first mother (in the cell labeled G) correspond to the parities of the Mohs ratings for the centroid G, for P, Q, R, S respectively. As 10, 3, 5, 7 are even, odd, odd, odd, this gives Caput Draconis.

After the four figures (called *mothers*) are placed, executing the geomancy procedure will produce the four *daughters*, and then similarly the rest of the chart. The completed chart is shown below, with the names of the geomantic figures annotated as well.

1212 Amissio | 1112 Cauda Draconis | 1121 Puer | 2211 Fortuna Major | 1122 Fortuna Minor | 1211 Puella | 2112 Conjunctio | 2111 Caput Draconis |

2122 Rubeus | 1112 Cauda Draconis | 2111 Caput Draconis | 2221 Tristitia | ||||

1212 Amissio | 2112 Conjunctio | ||||||

1122 Fortuna Minor |

Finally, take the numbers from the third step and index them into the name of the figures in the present step. Then read the chart from left to right, top to bottom:

Row | Figure position | Index | Parities | Symbol name | Letter |
---|---|---|---|---|---|

Row 1 | Fourth daughter | 3 | 1212 | Amissio | I |

Row 1 | Third daughter | 11 | 1112 | Cauda Draconis | N |

Row 1 | Second daughter | 1 | 1121 | Puer | P |

Row 1 | First daughter | 5 | 2211 | Fortuna Major | U |

Row 1 | Fourth mother (K) | 4 | 1122 | Fortuna Minor | T |

Row 1 | Third mother (I) | 6 | 1211 | Puella | A |

Row 1 | Second mother (H) | 6 | 2112 | Conjunctio | N |

Row 1 | First mother (G) | 13 | 2111 | Caput Draconis | S |

Row 2 | Fourth niece | 3 | 2122 | Rubeus | B |

Row 2 | Third niece | 7 | 1112 | Cauda Draconis | R |

Row 2 | Second niece | 12 | 2111 | Caput Draconis | I |

Row 2 | First niece | 1 | 2221 | Trisitia | T |

Row 3 | Left witness | 1 | 1212 | Amissio | A |

Row 3 | Right witness | 9 | 2112 | Conjunctio | I |

Row 4 | Judge | 6 | 1122 | Fortuna Minor | N |

This reads out INPUT ANS BRITAIN. The answer is **BRITAIN**.