Inflection point, so the statement [ a, a, a] holds, i.e., if that point is self-tangential. Lemma 1. If PF-06873600 Autophagy points a and b are inflection points and when the statement [ a, b, c] holds, then point c can also be an inflection point. Proof. The proof follows by applying the table a a a b b b c c . cExample 1. To get a additional visual representation of Lemma 1, consider the TSM-quasigroup given by the Cayley table a b c a a c b b c b a c b a c Lemma 2. If inflection point a may be the MCC950 Inhibitor tangential point of point b, then a and b are corresponding points. Proof. Point a would be the common tangential of points a and b. Instance 2. To get a much more visual representation of Lemma 2, look at the TSM-quasigroup provided by the Cayley table a b c d a a b d c b b a c d c d c b a d c d a b Proposition 1. If a and b will be the tangentials of points a and b, respectively, and if c is an inflection point, then [ a, b, c] implies [ a , b , c].Mathematics 2021, 9,3 ofProof. Based on [3] (Th. 2.1), [ a, b, c] implies [ a , b , c ], where c is definitely the tangential of c. Having said that, in our case c = c. Lemma three. If a and b would be the tangentials of points a and b respectively, and if [ a, b, c] and [ a , b , c], then c is an inflection point. Proof. The statement is followed by applying the table a a a b b b c c . cExample three. For a much more visual representation of Proposition 1 and Lemma 3, consider the TSMquasigroup offered by the Cayley table a b c d e a d c b a e b c e a d b c b a c e d d a d e b c e e b d c aLemma 4. If a and b will be the tangentials of points a and b, respectively, and if c is an inflection point, then [ a, b, d] and [ a , b , c] imply that c and d are corresponding points. Proof. From the table a a a b b b d d cit follows that point d has the tangential c, which itself is self-tangential. Example 4. For a more visual representation of Lemma 4, take into account the TSM-quasigroup given by the Cayley table a b c d e f g h a e d g b a h c f b d f h a g b e c c g h c d f e a b d b a d c e f h g e a g f e d c b h f h b e f c d g a g c e a h b g f d h f c b g h a d e Lemma 5. If the corresponding points a1 , a2 , and their prevalent second tangential a satisfy [ a1 , a2 , a ], then a is an inflection point. Proof. The statement follows on in the table a1 a1 a a2 a2 a a a awhere a is definitely the typical tangential of points a1 and a2 .Mathematics 2021, 9,four ofExample 5. For any more visual representation of Lemma five, think about the TSM-quasigroup given by the Cayley table a1 a2 a3 a4 a1 a3 a4 a1 a2 a2 a4 a3 a2 a1 a3 a1 a2 a4 a3 a4 a2 a1 a3 a4 Lemma 6. Let a1 , a2 , and a3 be pairwise corresponding points using the frequent tangential a , such that [ a1 , a2 , a3 ]. Then, a is definitely an inflection point. Proof. The proof follows from the table a1 a2 a3 a1 a2 a3 a a a.Instance six. For any more visual representation of Lemma 6, contemplate the TSM-quasigroup provided by the Cayley table a1 a2 a3 a4 a1 a4 a3 a2 a1 a2 a3 a4 a1 a2 a3 a2 a1 a4 a3 a4 a1 a2 a3 a4 Corollary 1. Let a1 , a2 , and a3 be pairwise corresponding points together with the popular tangential a , which is not an inflection point. Then, [ a1 , a2 , a3 ] does not hold. Lemma 7. Let [b, c, d], [ a, b, e], [ a, c, f ], and [ a, d, g]. Point a is definitely an inflection point if and only if [e, f , g]. Proof. Every single from the if and only if statements stick to on from one of the respective tables: b c d e f g a a a a a a b c d e f . gExample 7. For any additional visual representation of Lemma 7, contemplate the TSM-quasigroup offered by the Cayley table a b c d e f g a a e f g b c d b e f d c a b g c f d g b e a c d g c.