Show that a triangle's perimeter is more than 10 times the radius of its incircle.

The following table shows the ratio of the perimeter of a triangle to the radius of the incircle, based on the sizes of the two base angles. It's in two parts to prevent widening the frame of the comment area.

       5       10       15       20       25       30       35       40
  5 91.79004 68.93094 61.35169 57.59348 55.36485 53.90223 52.87866 52.13091
 10 68.93094 46.07286 38.49500 34.73857 32.51212 31.05215 30.03172 29.28767
 15 61.35169 38.49500 30.91891 27.16467 24.94087 23.48404 22.46731 21.72760
 20 57.59348 34.73857 27.16467 23.41307 21.19241 19.73928 18.72689 17.99222
 25 55.36485 32.51212 24.94087 21.19241 18.97545 17.52665 16.51931 15.79051
 30 53.90223 31.05215 23.48404 19.73928 17.52665 16.08290 15.08143 14.35947
 35 52.87866 30.03172 22.46731 18.72689 16.51931 15.08143 14.08679 13.37280
 40 52.13091 29.28767 21.72760 17.99222 15.79051 14.35947 13.37280 12.66811
 45 51.56857 28.72967 21.17464 17.44513 15.25026 13.82718 12.84982 12.15604
 50 51.13768 28.30382 20.75466 17.03199 14.84508 13.43131 12.46487 11.78397
 55 50.80420 27.97621 20.43389 16.71918 14.54158 13.13873 12.18515 11.51954
 60 50.54577 27.72462 20.19026 16.48486 14.31818 12.92820 11.98991 11.34256
 65 50.34732 27.53413 20.00908 16.31460 14.16079 12.78609 11.86607 11.24078
 70 50.19848 27.39460 19.88047 16.19886 14.06033 12.70390 11.80594 11.20755
 75 50.09218 27.29922 19.79796 16.13163 14.01137 12.67700 11.80594 11.24078
 80 50.02370 27.24361 19.75763 16.10958 14.01137 12.70390 11.86607 11.34256
 85 49.99015 27.22534 19.75763 16.13163 14.06033 12.78609 11.98991 11.51954
 90 49.99015 27.24361 19.79796 16.19886 14.16079 12.92820 12.18515 11.78397
 95 50.02370 27.29922 19.88047 16.31460 14.31818 13.13873 12.46487 12.15604
100 50.09218 27.39460 20.00908 16.48486 14.54158 13.43131 12.84982 12.66811
105 50.19848 27.53413 20.19026 16.71918 14.84508 13.82718 13.37280 13.37280
110 50.34732 27.72462 20.43389 17.03199 15.25026 14.35947 14.08679 14.35947
115 50.54577 27.97621 20.75466 17.44513 15.79051 15.08143 15.08143 15.79051
120 50.80420 28.30382 21.17464 17.99222 16.51931 16.08290 16.51931 17.99222
125 51.13768 28.72967 21.72760 18.72689 17.52665 17.52665 18.72689 21.72760
130 51.56857 29.28767 22.46731 19.73928 18.97545 19.73928 22.46731 29.28767
135 52.13091 30.03172 23.48404 21.19241 21.19241 23.48404 30.03172 52.13091
140 52.87866 31.05215 24.94087 23.41307 24.94087 31.05215 52.87866
145 53.90223 32.51212 27.16467 27.16467 32.51212 53.90223
150 55.36485 34.73857 30.91891 34.73857 55.36485
155 57.59348 38.49500 38.49500 57.59348
160 61.35169 46.07286 61.35169
165 68.93094 68.93094
170 91.79004
175

      45       50       55       60       65       70       75       80
  5 51.56857 51.13768 50.80420 50.54577 50.34732 50.19848 50.09218 50.02370
 10 28.72967 28.30382 27.97621 27.72462 27.53413 27.39460 27.29922 27.24361
 15 21.17464 20.75466 20.43389 20.19026 20.00908 19.88047 19.79796 19.75763
 20 17.44513 17.03199 16.71918 16.48486 16.31460 16.19886 16.13163 16.10958
 25 15.25026 14.84508 14.54158 14.31818 14.16079 14.06033 14.01137 14.01137
 30 13.82718 13.43131 13.13873 12.92820 12.78609 12.70390 12.67700 12.70390
 35 12.84982 12.46487 12.18515 11.98991 11.86607 11.80594 11.80594 11.86607
 40 12.15604 11.78397 11.51954 11.34256 11.24078 11.20755 11.24078 11.34256
 45 11.65685 11.30006 11.05390 10.89898 10.82409 10.82409 10.89898 11.05390
 50 11.30006 10.96153 10.73743 10.60941 10.56776 10.60941 10.73743 10.96153
 55 11.05390 10.73743 10.54022 10.44544 10.44544 10.54022 10.73743 11.05390
 60 10.89898 10.60941 10.44544 10.39230 10.44544 10.60941 10.89898 11.34256
 65 10.82409 10.56776 10.44544 10.44544 10.56776 10.82409 11.24078 11.86607
 70 10.82409 10.60941 10.54022 10.60941 10.82409 11.20755 11.80594 12.70390
 75 10.89898 10.73743 10.73743 10.89898 11.24078 11.80594 12.67700 14.01137
 80 11.05390 10.96153 11.05390 11.34256 11.86607 12.70390 14.01137 16.10958
 85 11.30006 11.30006 11.51954 11.98991 12.78609 14.06033 16.13163 19.75763
 90 11.65685 11.78397 12.18515 12.92820 14.16079 16.19886 19.79796 27.24361
 95 12.15604 12.46487 13.13873 14.31818 16.31460 19.88047 27.29922 50.02370
100 12.84982 13.43131 14.54158 16.48486 20.00908 27.39460 50.09218
105 13.82718 14.84508 16.71918 20.19026 27.53413 50.19848
110 15.25026 17.03199 20.43389 27.72462 50.34732
115 17.44513 20.75466 27.97621 50.54577
120 21.17464 28.30382 50.80420
125 28.72967 51.13768
130 51.56857
135

 

It shows a minimum when both base angles are 60 degrees--that is, when the triangle is equilateral.  That minimum is 10.39230, which is over 10.

Working out the value for an equilateral triangle, the perimeter is 6*sqrt(3) times the radius of the incircle, or about 10.39230484541326.

The program uses a triangle with base 1. The base angles are A and B. The third angle is C, and the angle subtended by AB at the incenter is Cp, standing for C'.

DEFDBL A-Z
pi = ATN(1) * 4: dr = pi / 180
CLS
PRINT "   ";
FOR A = 5 TO 40 STEP 5
 PRINT USING "  ###    "; A;
NEXT
PRINT

FOR B = 5 TO 175 STEP 5
PRINT USING "###"; B;
FOR A = 5 TO 40 STEP 5
  B2 = B / 2: A2 = A / 2
  Cp = 180 - A2 - B2
  BCp = SIN(A2 * dr) / SIN(Cp * dr)
  r = BCp * SIN(B2 * dr)
  C = 180 - A - B
  IF C > 0 THEN
    BC = SIN(A * dr) / SIN(C * dr)
    AC = SIN(B * dr) / SIN(C * dr)
    p = (1 + AC + BC)
    PRINT USING " ##.#####"; p / r;
  END IF
NEXT
  PRINT
NEXT

DO: LOOP UNTIL INKEY$ > ""

CLS
PRINT "   ";
FOR A = 45 TO 80 STEP 5
 PRINT USING "  ###    "; A;
NEXT
PRINT

FOR B = 5 TO 175 STEP 5
PRINT USING "###"; B;
FOR A = 45 TO 80 STEP 5
  B2 = B / 2: A2 = A / 2
  Cp = 180 - A2 - B2
  BCp = SIN(A2 * dr) / SIN(Cp * dr)
  r = BCp * SIN(B2 * dr)
  C = 180 - A - B
  IF C > 0 THEN
    BC = SIN(A * dr) / SIN(C * dr)
    AC = SIN(B * dr) / SIN(C * dr)
    p = (1 + AC + BC)
    IF B >= B THEN
     PRINT USING " ##.#####"; p / r;
    END IF
  END IF
NEXT
  PRINT
NEXT

 

Posted by Charlie on 2008-02-26 12:17:43