Proving Concurrency by Loci

A fascinating and catchy method for proving that a number of special lines are concurrent is using the concept of locus. This is now the classical method for proving the concurrency of the internal angle bisectors and perpendicular side bisectors of a triangle. In this paper, we prove the concurrency of the altitudes and the medians by showing that they are loci of some interesting points. Our proofs for these ancient theorems seem to be new. We also provide loci method proofs for the concurrency theorems of Ceva and Carnot .


Introduction.
A marvelous observation in the geometry of triangles is the fact that there are some special lines in triangles that are concurrent (i.e., the lines meet at one point). The proofs of some of these theorems (specially that of the concurrency of altitudes and medians) could be forgotten if one has been away from geometry for long.
Yet, a proof via loci (this is the Latin plural of locus, the location of all the points that share a certain property) may be easier to remember, even after many years. An example, maybe the simplest one, is recalling that a perpendicular side bisector of BC in △ABC is the locus of all the points X such that |XB| = |XC|. This means that a point X lies on the perpendicular side bisector of BC if and only if X has equal distances from the points B and C. Similarly, an internal angle bisector of ∠A is the locus of all the points X inside △ABC such that X has equal distances from the sides AB and AC. Thus, the three internal angle bisectors as well as the three perpendicular side bisectors of every triangle are concurrent. These theorems are proved in Euclid's Elements (as Propositions 4 and 5 of the book IV); see [1, §2] or [3, § §4.3]. The concurrency of the altitudes and the medians do not appear in the Elements of Euclid, though they are classical theorems by now. Some believe that Archimedes knew the concurrency of the medians (see [3, p. 84]), and two proofs for In its proof, we have considered the case where the altitude lies inside the triangle (cf. [1,Lemma]); other cases can be dealt with similarly.

Theorem 2.1 (Each Altitude is a Locus):
The locus of all the points X such that |XB| 2 −|XC| 2 = |AB| 2 −|AC| 2 is (the extended line of) the altitude AH (see Figure 1).

Proof:
If X is on the altitude AH, then apply Pythagoras' theorem to the four right triangles △ABH, △AHC, △XBH, and △XHC as follows: Now, suppose that we have |XB| 2 −|XC| 2 = |AB| 2 −|AC| 2 ; draw a perpendicular line from X to BC and assume that it meets BC at Y . By Pythagoras' theorem, Since for a similar reason we have |AB| 2 −|AC| 2 = |BC|·(|BH|−|HC|), from the presumed assumption  The altitudes of a triangle are concurrent.

Proof:
If X is the intersection of the altitudes drawn from B and C in △ABC, then by Theorem 2.1 we have Thus, |XB| 2 −|XC| 2 = |AB| 2 −|AC| 2 , which results by adding the two sides of the above equations. So, by Theorem 2.1, X lies on the altitude drawn from A too.

QED
The above proof does not appear among the 12 proofs for the concurrency of the altitudes listed in [1].
The concurrency of the altitudes (Corollary 2.2) as well as the concurrency of the perpendicular side bisectors are two special cases of Carnot's Concurrency Theorem which can be proved by using loci as follows.

Theorem 2.3 (Each Perpendicular is a Locus):
Let H be a point on and inside the line segment BC. The locus of all the points X such that |XB| 2 −|XC| 2 = |BH| 2 −|HC| 2 is the line perpendicular to BC at H (see Figure 1).

Proof:
If X is on the line perpendicular to BC with foot H, then we showed in the proof of Theorem 2.1 that we have then assume that the line perpendicular to BC from X meets BC at Y . Therefore, similar to the proof of Proof: Let X be the intersection of the perpendiculars to AB and AC with feet C ′ and B ′ , respectively. By Theo- Thus, by adding the two sides of these equations we get QED It is easy to see that Proof #7 in [1] actually infers the concurrence of the altitudes from Carnot's concurrency theorem. As a matter of fact, the altitudes are loci of some other, trigonometric, kind.  The locus of all the points X inside △ABC such that △AXB and △AXC have equal areas is the median AM (see Figure 2, by taking A ′ = M ).

Proof:
If

Corollary 3.2 (Medians Concur):
The medians of a triangle are concurrent.

Proof:
If X is the intersection of the medians drawn from B and C in △ABC, then by Theorem 3.1 we have S △BXA = S △BXC and S △CXA = S △CXB . Thus, S △AXB = S △AXC , and so by Theorem 3.1, X lies on the median drawn from A too.

QED
It is now known that the concurrency of the internal angle bisectors, altitudes (Corollary 2.2), and medians (Corollary 3.2) are special cases of Ceva's Concurrency Theorem, which can also be proved by using loci. Let us recall that a Cevian is a line segment that connects a vertex of a triangle to a point on the opposite side.

Theorem 3.3 (Each Cevian is a Locus):
Let A ′ be a point on and inside BC. The Cevian AA ′ is the locus of all the points X inside △ABC such that S △AXB S △AXC = |BA ′ | |A ′ C| (see Figure 2).

Proof:
Suppose, first, that X lies on AA ′ . Then we have

Corollary 3.4 (Ceva's Concurrency Theorem):
Let A ′ , B ′ , and C ′ be on (and inside), respectively, the sides BC, AC, and AB of △ABC. The Cevians AA ′ , BB ′ and CC ′ are concurrent if and only if Ceva's identity holds:

Proof:
If X is the intersection of BB ′ and CC ′ , then by Theorem 3.3 we have Thus, by multiplying the two sides of these equations we get   by (1) and (2).
So, by Proposition 3.5, the point X lies on the median through A too. shows that an internal bisector of an angle concurs with the external bisectors of the two other angles. The perpendicular side bisectors of a triangle concur because each of them is the locus of all the points that are equidistant from the two vertices of the side. These two theorems appear in Euclid's Elements, and are proved in that book by using loci. Two other now-classical concurrency theorems, that of the altitudes and the medians, do not appear there; though, the ancient Greeks had all the tools for proving those theorems. Could a reason for this exclusion be that no proof by using loci was know for them? In this paper, we proved these two theorems by the loci method. We noted that the altitude AH of △ABC is the locus of all the points X on the plane such that |XB| 2 − |XC| 2 is the fixed value |AB| 2 − |AC| 2 (Theorem 2.1); and the median AM is the locus of all the points X inside △ABC such that the triangles △AXB and △AXC have equal areas (Theorem 3.1). Thus, we presented proofs for the concurrence of the altitudes (Corollary 2.2) and the medians (Corollary 3.2) by using loci. As a generalization, we showed (Theorem 2.3) that a perpendicular line to BC from a point A ′ on it is the locus of all the points X such that |XB| 2 −|XC| 2 is the fixed value |BA ′ | 2 −|A ′ C| 2 .