Relations between various elements of a triangle from Interactive Mathematics Miscellany and Puzzles

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Relations between various elements of a triangle

S = rp

Triangle ABC is a union of three triangles ABI, BCI, CAI, with bases AB = c, BC = a, and AC = b, respectively. The altitudes to those bases all have the length of r.

r² = p^-1(p - a)(p - b)(p - c)

This follows from S² = p(p - a)(p - b)(p - c) and S = rp.

1/r = 1/h_a + 1/h_b + 1/h_c

2S = ah_a = bh_b = ch_c. Therefore, a = 2S/h_a, etc. On the other hand, S = rp, so that p = S/r, or (a + b + c) = 2S/r. From here, 2S/h_a + 2S/h_b + 2S/h_c = 2S/r.

sin²(A/2) = (p - b)(p - c) / bc, etc.

First of all, sin(A) = 2·sin(A/2)cos(A/2) = 2·sin²(A/2)/tan(A/2). Therefore,

(1)	sin²(A/2) = sin(A)·tan(A/2) /2.

We know that

(2)	sin(A) = 2S / bc

and

(3)	tan(A/2) = r/(p - a).

Combining (1)-(3) gives

sin²(A/2) = 2S/bc · r/(p-a) · 1/2.

Taking into account that S² = p(p - a)(p - b)(p - c) and r² = p^-1(p - a)(p - b)(p - c), the latter leads to

sin²(A/2) = (p - b)(p - c) / bc.

AI² = (p - a)bc/p

Square the obvious

AI = r/sin(A/2).

Substitute there sin²(A/2) = (p - b)(p - c) / bc and r² = p^-1(p - a)(p - b)(p - c):

AI² = p^-1(p - a)(p - b)(p - c)bc/(p - b)(p - c)

= (p - a)bc/p.

bc·tan(B/2)·tan(C/2)

Squaring AI = r/sin(A/2) and substituting sin²(A/2) = (p - b)(p - c) / bc, we obtain

AI² = r²·bc/(p - b)(p - c).

By the incenter construction, tan(B/2) = r/(p - b) and also tan(C/2) = r/(p - c). Substituting these into the above gives the required

AI² = bc·tan(B/2)·tan(C/2).

1/r = 1/r_a + 1/r_b + 1/r_c

As we know,

S = r_a(p - a) = r_b(p - b) = r_c(p - c).

Therefore

1/r_a+ 1/r_b + 1/r_c	= (p - a)/S + (p - b)/S + (p - c)/S
	= (3p - a - b - c)/S
	= (3p - 2p)/S
	= p/S
	= 1/r,

since S = rp.

r_a + r_b + r_c = r + 4R

As we know,

S = rp,

and also

S = r_a(p - a) = r_b(p - b) = r_c(p - c).

From these we have

(4)	r_a + r_b + r_c - r = S(1/(p - a) + 1/(p - b) + 1/(p - c) - 1/p).

Simple algebra yields

1/(p - a) + 1/(p - b) = c / (p - a)(p - b) and
1/(p - c) - 1/p = c / p(p - c).

And a little more effort makes a great payoff:

c / (p - a)(p - b) + c / p(p - c) = abc / p(p - a)(p - b)(p - c) = abc / S²,

by Heron's formula. To sum up, from (4)

(5)	r_a + r_b + r_c - r = S·abc/S² = abc / S.

However, abc = 4RS, so that (5) implies exactly what's needed:

r_a + r_b + r_c - r = abc / S = 4R.

r_ar_br_c = pS

Since

S = r_a(p - a) = r_b(p - b) = r_c(p - c),

we immediatly obtain

r_ar_br_c	= S³ / (p - a)(p - b)(p - c)
	= S³ / [S² / p],

by Heron's formula. But

S³ / [S² / p] = Sp.

r r_ar_br_c = S²

This is an immediate consequence of r_ar_br_c = pS and rp = S.

m_a² = (b² + c²)/2 - a²/4

Let's use Stewart's theorem

AB²·DC + AC²·BD - AD²·BC = BC·DC·BD

with D being the midpoint M of BC. Then AB = c, DC = a/2, AC = b, BD = a/2, AD = m_a, BC = a. We have,

c²·a/2 + b²·a/2 - m_a²·a = a·a/2·a/2.

(The above identity could be as easily obtained with the help of the theorem of Cosines.)

abc = 4RS

Let AD be a diameter of the circumcircle of ABC and AH its altitude. Right triangles AHC and ABD are similar, for ADB = ACH. Therefore,

AH/AC = AB/AD.

In other words,

2R·AH = AB·AC = bc.

And finally

abc = 2R·AH·a = 4RS.

bc = 2Rh_a

This follows from the previous derivation or by substituting S = ah_a/2 into the final formula.

p = 4Rcos(A/2)·cos(B/2)·cos(C/2)

By the Law of Sines

a = 2R·sinA, b = 2R·sinB, c = 2R·sinC,

so that

p	= R·(sinA + sinB + sinC)
	= R·(sinA + sinB + sin(180° - A - B)
	= R·(sinA + sinB + sin(A + B)
	= R·(sinA + sinB + sinA·cosB + cosA·sinB)
	= R·(sinA·(1 + cosB) + sinB·(1 + cosA))
	= R·(2sin(A/2)cos(A/2)·2cos²(B/2) + 2sin(B/2)cos(B/2)·2cos²(A/2))
	= 4R·cos(A/2)cos(B/2)(sin(A/2)cos(B/2) + sin(B/2)cos(A/2))
	= 4R·cos(A/2)cos(B/2)sin((A + B)/2)
	= 4R·cos(A/2)cos(B/2)sin(90° - C/2)
	= 4R·cos(A/2)cos(B/2)cos(C/2).

S = 2R²sin(A)·sin(B)·sin(C)

By the Law of Sines

a = 2R·sinA, b = 2R·sinB,

For the area of the triangle we have

2S	= ab·sinC
	= 2RsinA·2RsinB·sinC
	= 4R²·sinA·sinB·sinC.

r = 4Rsin(A/2)·sin(B/2)·sin(C/2)

This follows directly from

cot(A/2) + cot(B/2) + cot(C/2) = cot(A/2)·cot(B/2)·cot(C/2)

This is equivalent to showing that, for A + B + C = 180°,

cos(A/2)sin(B/2)sin(C/2) + sin(A/2)cos(B/2)sin(C/2) + sin(A/2)sin(B/2)cos(C/2)
= cos(A/2)cos(B/2)cos(C/2).

Let's transform the left-hand side:

cos(A/2)sin(B/2)sin(C/2) + sin(A/2)cos(B/2)sin(C/2) + sin(A/2)sin(B/2)cos(C/2)
= sin((A+B)/2)sin(C/2) + sin(A/2)sin(B/2)cos(C/2).

But since (A + B)/2 C = 90° - C/2, this equals

cos(C/2)sin(C/2) + sin(A/2)sin(B/2)cos(C/2) = cos(C/2)[sin(C/2) + sin(A/2)sin(B/2)].

Reversing the steps:

sin(C/2) + sin(A/2)sin(B/2)	= cos((A+B)/2) + sin(A/2)sin(B/2)
	= cos(A/2)cos(B/2) - sin(A/2)sin(B/2) + sin(A/2)sin(B/2)
	= cos(A/2)cos(B/2).

Combining everything together we get the desired identity.

rR = abc / 4p

r² = p^-1(p - a)(p - b)(p - c) is equivalent to

r = √D / p,

where D = p(p - a)(p - b)(p - c). Also,

R = abc / 4√D.

Multiplying the two gives

rR = abc / 4p.

Note that the identity at hand also follows by combining S = rp with abc = 4RS.

AH = 2R·|cos(A)|

In ΔABH, if A < 90°, ∠ABH = 90° - ∠A. (This is because ΔABH_b is right.) Applying the law of sines to ΔABH gives,

	AH / sin(∠ABH)	= AB / sin(180° - ∠C)
		= AB / sin(∠C)
		= 2R

from the lawa of sines applied in ΔABC. Thus

	2R	= AH / sin(∠ABH)
		= AH / sin(90° - ∠A)
		= AH / cos(∠A),

which proves the assertion AH = 2R·|cos(A)| when A < 90°.

For the case where ∠A is obtuse, H falls outside ΔABC, ∠ABH = ∠A - 90° so at the end we'll get AH = -2R·cos(A), proving AH = 2R·|cos(A)| in this case also.