INTERNATIONAL JOURNAL OF GEOMETRY Vol. 10 (2021), No. 3, 21 - 47 MORE NEW CHARACTERIZATIONS OF TANGENTIAL QUADRILATERALS MARTIN JOSEFSSON and MARIO DALC´IN Abstract. This is a continuation to the investigations we made in [15] about tangential quadrilaterals. Here we prove 24 more necessary and suf- ficient conditions for when a convex quadrilateral can have an incircle. 1. Introduction A convex quadrilateral in which a circle can be drawn that is internally tangent to all four sides is most often called a tangential quadrilateral or a circumscribed quadrilateral, but several other similar names have been used (see [6, p. 65]). This type of quadrilateral was first considered in the thirteenth century by Jordanus Nemorarius according to [1, p. 300], but it is unclear what discoveries he made. The most important and fundamental theorem for a tangential quadrilateral ABCD is called Pitot’s theorem after the French engineer Henri Pitot, who in 1725 proved that (1) AB + CD = BC + DA. This condition is also sufficient, which was proved by J. B. Durrande in 1815 [18, p. 64]. At the time of writing this paper, more than ten different proofs of this converse are known (most of these references are given in [14]). Just over a decade ago, the interest in these quadrilaterals was greatly increased, resulting in a large number of necessary and sufficient conditions being discovered and proved. The papers [16, 6, 8, 9, 10, 15] contain al- together 49 different characterizations (either proved or reviewed), and the present paper is a continuation to the latter of these. Three other were proved in [2], [5, p. 395] and [12, p. 148]. Adding the 24 necessary and sufficient conditions in this paper results in a total of 76, and makes the tangential quadrilateral the undisputed leader on the list of objects in ge- ometry with the greatest number of known characterizations (the cyclic quadrilateral is on second place with 47 published characterizations). ————————————– Keywords and phrases: Tangential quadrilateral, Pitot’s theorem, cyclic quadrilateral, concurrent lines, Newton’s line (2020)Mathematics Subject Classification: 51M04 Received: 07.10.2020. In revised form: 06.11.2020. Accepted: 02.11.2020. 22 Martin Josefsson and Mario Dalc´ın 2. Variations on an old theorem In this section we shall study three reformulations of Pitot’s theorem. We call a circle tangent to one side of a quadrilateral and the extensions of the adjacent two sides an escribed circle, as was done in [8, p. 71]. Theorem 2.1. In a convex quadrilateral, the sum of the four distances along the extensions of opposite sides between each vertex and the tangency point of the adjacent escribed circle are equal for both pairs of opposite sides if and only if it is a tangential quadrilateral. Figure 1. ABCD is tangential ⇔ equality (2) holds Proof. Using notations as in Figure 1, we have according to the two tangent theorem that AE = AQ, EB = BL and so on. By Pitot’s theorem, ABCD is tangential if and only if AE + EB + CG + GD = BF + FC + DH + HA which is equivalent to (2) AQ + BL + CM + DP = BK + CN + DO + AJ. This last equality is what the theorem states. The following characterization is related to Theorem 3.2 in [15]. Theorem 2.2. In a convex quadrilateral there are four excircles to the four overlapping triangles created by the diagonals that are tangent to the diago- nals. The sum of the four distances between the vertices and the tangency points on the extensions of two opposite sides are equal to the sum of the four distances between the vertices and the tangency points on the extensions of the other pair of opposite sides if and only if it is a tangential quadrilateral. Proof. In [15, p. 57], we proved that in all convex quadrilaterals, ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ BC +DA−CD −AB = 2(S Z −X Y )= S Z +V W −X Y −U T More new characterizations of tangential quadrilaterals 23 Figure 2. ABCD is tangential ⇔ equality (3) holds where the last equality is due to the fact that S′′Z′′ = V ′′W ′′ and X′′Y ′′ = U ′′T ′′, also according to [15, p. 57] (see Figure 2). Thus we get ′′ ′′ ′′ ′′ BC + DA − CD − AB = S A + AB + BZ + V D + DC + CW − X′′A − AD − DY ′′ − U ′′B − BC − CT ′′ which is simplified as ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ 2(BC+DA−CD−AB)= S A+BZ +V D+CW −X A−DY −U B−CT . Hence BC + DA = CD + AB is equivalent to ′′ ′′ ′′ ′′ ′′ ′′ ′′ ′′ (3) AS + BZ + CW + DV = AX + BU + CT + DY which by Pitot’s theorem finishes the proof. Next we have a generalization of Theorem 2 in [6]. Two special cases of the ‘if’ part of this characterization were proved as Theorems 5.4.1 and 5.4.2 in [17, p. 150]. Theorem 2.3. Related to a convex quadrilateral ABCD, there is an arbi- trary circle tangent to the lines AB and BC at S and T , an arbitrary circle tangent to the lines BC and CD at U and V , an arbitrary circle tangent to the lines CD and DA at W and X, and an arbitrary circle tangent to the lines DA and AB at Y and Z respectively. The relation ± ZS ± VW = ± TU ± XY holds if and only if it is a tangential quadrilateral, where the minus sign for each term is chosen if the tangency points have reversed alphabetical order on each line when tracing around the quadrilateral in counter clockwise direction starting at S, given that ABCD is labeled in counter clockwise direction. Proof. There are several possible cases with internal, external, or partially external circles. We start with the simple case in Figure 3. In a convex 24 Martin Josefsson and Mario Dalc´ın Figure 3. ABCD is tangential ⇔ ZS + VW = T U + XY quadrilateral ABCD with only internal circles, we have AB − BC + CD − DA = AZ + ZS + SB − BT − TU − UC + CV + VW + WD − DX − XY − Y A = ZS − TU + VW − XY according to the two tangent theorem (AZ = Y A and so on). The quadri- lateral is tangential if and only if AB +CD = BC +DA, which is equivalent to (4) ZS + VW = TU + XY. Figure 4. In this case, ABCD is tangential ⇔ ZS + VW = −T U + XY In a more general case as the one in Figure 4, it holds that AB − BC + CD − DA = − AZ ± ZS + SB − (BT ± TU + UC) + CV ± VW + WD − (DX ± XY − Y A) = ± ZS ∓ TU ± VW ∓ XY where the plus signs are chosen when the tangency points on a (possibly extended) side of the quadrilateral are in alphabetical order, otherwise minus More new characterizations of tangential quadrilaterals 25 signs are chosen. Also, the tangent lengths at an external circle are negative (AZ and Y A in Figure 4). ABCD is a tangential quadrilateral if and only if AB + CD = BC + DA ⇔ ± ZS ± VW = ±TU ± XY. In Figure 5, we have drawn an example where all tangency points are in reversed alphabetical order, making all four terms negative. By shrinking the smallest circle, we can easily get the case when there are two negative terms. We let the reader draw the case with three negative terms. Figure 5. ABCD is tangential ⇔ −ZS − VW = −T U − XY It may seem amazing that there is no size constraint on the four circles. But an increase or decrease of any circle add or subtract the same length to both sides of equality (4). 3. Other subtriangle lengths ′ ′ ′ ′ First we have a variant of the true characterization T1T3 = T2T4 of tan- gential quadrilaterals that appeared in the proof of Theorem 3 in [6]. (Note ′ ′ ′ ′ that the equality T1T2 = T3T4 that was stated in that theorem is not a valid characterization of tangential quadrilaterals, since it is also true in paral- lelograms. The first equality however is case-independent and thus remains true no matter where the subtriangle incircles are tangent to the diagonals, which the second equality is not.) Theorem 3.1. In a convex quadrilateral ABCD where the diagonals in- tersect at P , the distance between the points where two opposite incircles in triangles ABP , BCP , CDP , DAP are tangent to one diagonal is equal to the distance between the tangency points for the other two opposite incircles on the other diagonal if and only if ABCD is a tangential quadrilateral. 26 Martin Josefsson and Mario Dalc´ın Figure 6. ABCD is tangential ⇔ T1X1 = V1Z1 Proof. In all convex quadrilaterals, it holds that U1P = V1P and Y1P = Z1P according to the two tangent theorem; whence U1Y1 = V1Z1 (see Fig- ure 6). In the proof of Theorem 3 in [6], it was proved that ABCD is tangential if and only if T1X1 = U1Y1 (but the origin of that theorem was a solution to a Russian geometry problem by I. Vaynshtejn, as noted in [6, p. 68]). Thus we get that T1X1 = V1Z1 is a characterization of tangential quadrilaterals. As a consequence we have the following theorem, where we use the nota- tion [S1T1W1X1] for the area of quadrilateral S1T1W1X1. Theorem 3.2. The four tangency points on the diagonals of a convex quadrilateral for two opposite non-overlapping subtriangle incircles are the vertices of an isosceles trapezoid. The two isosceles trapezoids created this way have equal area if and only if the original quadrilateral is tangential. Figure 7. ABCD is tangential ⇔ [S1T1W1X1]=[Y1Z1U1V1] More new characterizations of tangential quadrilaterals 27 Proof. We conclude that opposite chords Y1Z1 and V1U1 are parallel by applying the two tangent theorem, the isosceles triangle theorem and equal alternating angles in quadrilateral Y1Z1U1V1 (see Figure 7). Since the two diagonals in that trapezoid are equal by the two tangent theorem, it is isosceles.