cp's OEIS Frontend

Theorem 1: Consider a triangle whose area A, sides (a,b,c), inradius r and the radius of whose three excircles r1, r2, r3 are integers. Then the sum a^2 + b^2 + c^2 + r^2 + r1^2 + r2^2 + r3^2 is a perfect square equal to 16R^2, where R is the circumradius.

Proof: (r1 + r2 + r3 - r)^2 = r1^2 + r2^2 + r3^2 + r^2 + a^2 + b^2 + c^2 because: r1*r2 + r2*r3 + r3*r1 - r*r1 - r*r2 - r*r3 = (a^2 + b^2 + c^2)/2 (formula from Feuerbach - see the link). But r1 + r2 + r3 - r = 4*R (see the reference: Johnson 1929, pp. 190-191), hence the result. Remark: R is not necessarily an integer; for example, at a(1) = 6 with (a,b,c) = (3, 4, 5) we obtain r = 1, r1 = 2, r2 = 3, r3 = 6 and R = 5/2. Then 3^2 + 4^2 + 5^2 + 1^2 + 2^2 + 3^2 + 6^2 = 16*(5/2)^2 = 10^2. Nevertheless, if R is an integer, then r, r1, r2 and r3 are necessarily integers (see the following theorem). The subset of a(n) with R integer is A208984 = {24, 96, 120, 168, 216, 240, 336, 384, 432, 480, 600, ...}

Theorem 2: Consider a triangle whose area A, sides (a,b,c) and circumradius R are integers. Then the inradius r and the radius of the three excircles r1, r2, r3 are also integers.

Proof: Let s be the semiperimeter, let s*A = r1*r2*r3 be an integer, and let r*r1*r2*r3 = A^2 also be an integer => r is an integer. r1 = A/(s-a), r2 = A/(s-b), r3 = A/(s-c) => r1*r2 = s*(s-c), r1*r3=s*(s-b), r2*r3 = s*(s-a) are integers. Because r1*r2*r3 is an integer => r1, r2, r3 are integers.

Subsequence of A231174. The sequence is probably infinite.

The corresponding sequence of the primes d is {5, 5, 13, 17, 53, 193, 241,...} (see A350379).

In geometry, Euler's theorem states that the distance between the incenter and circumcenter can be expressed as d = sqrt(R(R-2r)), where R is the circumradius and r is the inradius.

The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2.

The inradius r is given by r = A/s and the circumradius is given by R = abc/4A.

The following table gives the first values (A, a, b, c, r, R, d) where A is the area of the triangles, a, b, c are the integer sides of the triangles, r is the inradius, R is the circumradius and d is the distance between the incenter and circumcenter with d = sqrt(R(R-2r)).

+------------+--------+--------+---------+---------+---------+-----+

| A | a | b | c | r | R | d |

+------------+--------+--------+---------+---------+---------+-----+

| 48 | 10 | 10 | 16 | 8/3 | 25/3 | 5 |

| 768 | 40 | 40 | 48 | 12 | 25 | 5 |

| 3840 | 80 | 104 | 104 | 80/3 | 169/3 | 13 |

| 108000 | 480 | 510 | 510 | 144 | 289 | 17 |

| 1134000 | 1590 | 1590 | 1680 | 1400/3 | 2809/3 | 53 |

| 200202240 | 21280 | 21616 | 21616 | 18620/3 | 37249/3 | 193 |

| 4382077920 | 100320 | 100738 | 100738 | 29040 | 58081 | 241 |

....................................................................

From the previous table, we observe that the triangles are isosceles, the distance between the incenter and the circumcenter is d = sqrt(R) if R is a perfect square, or d = sqrt(3R) if R is of the form k^2/3, k integer. We also observe that d divides the two equal sides of the isosceles triangles: 10/5 = 2, 40/5 = 8, 104/13 = 8, 510/17 = 30, 1590/853 = 30, 21616/193 = 112, 100738/241 = 418, ....

The corresponding integer areas of integer-sided triangles such that the distance between the incenter and the circumcenter is a prime number is given by the sequence A350378.

In geometry, Euler's theorem states that the distance between the incenter and circumcenter can be expressed as d = sqrt(R(R-2r)), where R is the circumradius and r is the inradius.

The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2.

The inradius r is given by r = A/s and the circumradius is given by R = abc/4A.

The following table gives the first values (A, a, b, c, r, R, d) where A is the area of the triangles, a, b, c are the integer sides of the triangles, r is the inradius, R is the circumradius and d is the distance between the incenter and circumcenter with d = sqrt(R(R-2r)).

+------------+--------+--------+---------+---------+---------+-----+

| A | a | b | c | r | R | d |

+------------+--------+--------+---------+---------+---------+-----+

| 48 | 10 | 10 | 16 | 8/3 | 25/3 | 5 |

| 768 | 40 | 40 | 48 | 12 | 25 | 5 |

| 3840 | 80 | 104 | 104 | 80/3 | 169/3 | 13 |

| 108000 | 480 | 510 | 510 | 144 | 289 | 17 |

| 1134000 | 1590 | 1590 | 1680 | 1400/3 | 2809/3 | 53 |

| 200202240 | 21280 | 21616 | 21616 | 18620/3 | 37249/3 | 193 |

| 4382077920 | 100320 | 100738 | 100738 | 29040 | 58081 | 241 |

....................................................................

From the previous table, we observe that the triangles are isosceles, the distance between the incenter and the circumcenter is d = sqrt(R) if R is a perfect square, or d = sqrt(3R) if R is of the form k^2/3, k integer. We also observe that d divides the two equal sides of the isosceles triangle: 10/5 = 2, 40/5 = 8, 104/13 = 8, 510/17 = 30, 1590/853 = 30, 21616/193 = 112, 100738/241 = 418, ....

For n <= 200, the number of distinct integer-sided triangles inscribed in a circle of radius A009003(n) whose inradius are integers belongs to the set E = {1, 5, 10, 12, 38} where a(168) = 38 (see the table given in reference). Is the set E infinite when n is infinite?

a(m) > 1 for m = 7, 18, 26, 31, 35, ... and {A009003(m)} = {25, 50, 65, 75, 85, ...} = {A009177}.

We observe geometric properties:

If a(n) = 1, the unique triangle is a right triangle.

If a(n) = 5, we find two right triangles, two isosceles triangles and another triangle (neither isosceles nor right triangle).

If a(n) = 10, we find three right triangles, two isosceles triangles and five other triangles.

If a(n) = 12, we find four right triangles and eight other triangles.

The area A of a triangle whose sides have lengths u, v, and w is given by Heron's formula: A = sqrt(s*(s-u)*(s-v)*(s-w)), where s = (u+v+w)/2.

The inradius r is given by r = A/s and the circumradius is given by R = u*v*w/4A.

The primitive areas are 168, 338, 432, 600, 768, 13608, 14280, 20280, 27216, ...

The non-primitive areas 16*a(n) are in the sequence because if R is the circumradius corresponding to a(n), then 4*R is the circumradius corresponding to 16*a(n).

Each circumradius belongs to the sequence {25, 100, 169, 225, 289, 400, 625, 676, ...}, and it seems that this last sequence is A198385 (second of a triple of squares in arithmetic progression).

The following table gives the first values (A, R, a, b, c) where A is the integer area, R the radius of the circumcircle, and a, b, c are the integer sides of the triangle.

**************************************

* A * R * a * b * c *

**************************************

* 168 * 25 * 14 * 30 * 40 *

* 336 * 25 * 14 * 48 * 50 *

* 432 * 25 * 30 * 30 * 48 *

* 600 * 25 * 30 * 40 * 50 *

* 768 * 25 * 40 * 40 * 48 *

* 2688 * 100 * 56 * 120 * 160 *

* 5376 * 100 * 56 * 192 * 200 *

* 6912 * 100 * 120 * 120 * 192 *

* 9600 * 100 * 120 * 160 * 200 *

* 12288 * 100 * 160 * 160 * 192 *

* 13608 * 225 * 126 * 270 * 360 *

* 14280 * 169 * 130 * 238 * 312 *

* 20280 * 169 * 130 * 312 * 338 *

* 27216 * 225 * 126 * 432 * 450 *

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