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1 | ;;
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2 | ;; Prove Apollonius Circle Theorem:
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3 | ;;----------------------------------------------------------------
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4 | ;; If ABC is a right triangle with hypotenuse BC,
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5 | ;; and
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6 | ;;
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7 | ;; 1) M is the midpoint of BC;
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8 | ;; 2) M1 is the midpoint of AB;
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9 | ;; 3) M2 is the midpoint of AC;
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10 | ;; 4) is the foot of the altitude dropped from A;
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11 | ;;
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12 | ;; then A, H, M1, M2 and M lie on the same circle.
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13 | ;;----------------------------------------------------------------
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14 | ;;
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15 |
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16 | (prove-theorem
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17 |
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18 | ;; If
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19 | (
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20 | (perpendicular A B A C) ; AB _|_ AC
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21 | (midpoint B C M) ; M is the midpoint of BC
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22 | (midpoint A M O) ; O is the midpoint of AM
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23 | (collinear B H C) ; H lies on BC
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24 | (perpendicular A H B C) ; AH _|_ BC
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25 | )
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26 | ;; Then
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27 | (
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28 | (equidistant M O H O) ; MO = HO
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29 | ;; Or
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30 | (identical-points B C) ; B = C
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31 | )
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32 | )
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