Unit III Summary: Circles, Projective Loci & Triangle Center Benchmarks¶
Quantitative Synopsis of Problems XXI & XXV–LXIII
1. Circle, Proportion & Projective Benchmarks¶
Prob. |
Construction |
Canon Ref. |
Classical (\(S\)) |
Lemoine Symbol (\(S\)) |
\(\Delta S\) |
|---|---|---|---|---|---|
XXI |
Circumcircle through \(A, B, C\) |
IV.5 · \(X(3)\) |
\(17\) (5 circles) |
\(4R_1+2R_2+5C_1+4C_3\) (\(15\)) |
\(-2\) |
XXVI |
External tangents to circle \(O\) |
III.17 |
\(19\) (4 lines) |
\(5R_1+3R_2+6C_1+4C_3\) (\(18\)) |
\(-1\) |
XXVII |
Incircle of \(\triangle ABC\) |
IV.4 · \(X(1)\) |
\(30\) (10 circles) |
\(4R_1+2R_2+11C_1+6C_3\) (\(23\)) |
\(-7\) |
XXXV |
Mean proportional \(\sqrt{ab}\) |
II.14 / VI.13 |
\(24\) (6 circles) |
\(2R_1+2R_2+6C_1+C_2+3C_3\) (\(14\)) |
\(-10\) |
XXXVI |
Golden section \(\phi\) |
II.11 / VI.30 |
\(25\) (3 lines) |
\(2R_1+R_2+9C_1+6C_3\) (\(18\)) |
\(-7\) |
XXXVIII |
Circle inversion (\(OA > R/2\)) |
III.36 |
\(12\) (2 lines) |
\(5C_1+3C_3\) (\(8\), 0 lines) |
\(-4\) |
XXXIX |
Polar of point w.r.t. circle |
Pappus VII |
\(25\) (4 circles) |
\(12R_1+7R_2\) (\(19\), 0 circles) |
\(-6\) |
XLI |
Radical axis of two circles |
III.35–36 |
\(58\) (Mannheim) |
\(10R_1+5R_2+2C_3\) (\(17\)) |
\(-41\) |
XLVI |
Harmonic conjugate \((A,B;C,D)=-1\) |
Pappus VII |
\(22\) (2 circles) |
\(9R_1+6R_2\) (\(15\), 0 circles) |
\(-7\) |
2. Encyclopedia of Triangle Centers (ETC) Complexity Spectrum¶
Prob. |
Triangle Center |
ETC Index |
Optimal Lemoine Symbol |
\(S\) |
\(E\) |
|---|---|---|---|---|---|
XXI |
Circumcenter \(O\) (center only: \(S=12\)) |
\(X(3)\) |
\(4R_1 + 2R_2 + 5C_1 + 4C_3\) (with circle) |
15 |
9 |
LXI |
Nagel Point |
\(X(8)\) |
\(4R_1 + 2R_2 + 7C_1 + 3C_3\) |
16 |
11 |
L |
Centroid \(G\) |
\(X(2)\) |
\(6R_1 + 3R_2 + 6C_1 + 2C_3\) |
17 |
12 |
XXVII |
Incenter \(I\) (with incircle) |
\(X(1)\) |
\(4R_1 + 2R_2 + 11C_1 + 6C_3\) |
23 |
15 |
LVI |
Steiner Point |
\(X(99)\) |
\(8R_1 + 4R_2 + 9C_1 + 3C_3\) |
24 |
17 |
LI |
Lemoine Symmedian Point \(K\) |
\(X(6)\) |
\(4R_1 + 2R_2 + 13C_1 + 8C_3\) |
27 |
17 |
LIX |
Gergonne Point |
\(X(7)\) |
\(10R_1 + 5R_2 + 9C_1 + 3C_3\) |
27 |
19 |
LIV |
Single Brocard Point \(\omega\) |
Bicentric |
\(8R_1 + 4R_2 + 10C_1 + 6C_3\) |
28 |
18 |
LVII |
Tarry Point |
\(X(98)\) |
\(10R_1 + 5R_2 + 8C_1 + 5C_3\) |
28 |
18 |
LX |
Spieker Center |
\(X(10)\) |
\(8R_1 + 4R_2 + 10C_1 + 7C_3\) |
29 |
18 |
LV |
Twin Brocard Pair \((\omega, \omega')\) |
Bicentric |
\(12R_1 + 6R_2 + 12C_1 + 8C_3\) |
38 |
24 |