99-Two conjectures in inequality

1, Let $\lambda_1+\lambda_2+...+\lambda_n=1$ , $\lambda_i>0$   for  $ i=1,2,..,n$; f,f' are real positive continuous function that is concave up in [a,b], If $x_1, . . . , x_n$ and $y_1, . . . , y_n$ are numbers in [a,b] such that $(x_1, . . . , x_n)$ majorizes $(y_1, . . . , y_n)$, then:

\[\frac{\lambda_1f(x_1)+\lambda_2f(x_2)+....+\lambda_nf(x_n)}{f({\lambda_1x_1+\lambda_2x_2+....+\lambda_nx_n})} \geq \frac{\lambda_1f(y_1)+\lambda_2f(y_2)+....+\lambda_nf(y_n)}{f({\lambda_1y_1+\lambda_2y_2+....+\lambda_ny_n})}\]

2, Let $\lambda_1+\lambda_2+...+\lambda_n=1$ , $\lambda_i>0$   for  $ i=1,2,..,n$; f,f' are  real continuous function that is concave up in [a,b], If $x_1, . . . , x_n$ and $y_1, . . . , y_n$ are numbers in [a,b] such that $(x_1, . . . , x_n)$ majorizes $(y_1, . . . , y_n)$, then:

$$\lambda_1f(x_1)+\lambda_2f(x_2)+....+\lambda_nf(x_n)-f({\lambda_1x_1+\lambda_2x_2+....+\lambda_nx_n})$$
$$\geq \lambda_1f(y_1)+\lambda_2f(y_2)+....+\lambda_nf(y_n)-f({\lambda_1y_1+\lambda_2y_2+....+\lambda_ny_n})$$

98-Around Anticenter

Problem 1: Let $ABCD$ be a quadrilateral, $E$ be the midpoint of $BC$, $F$ be the midpoint of $AD$. The line though $E$ and perpendicular to $BC$ meets the line through $F$ and perpendicular $AD$ at $G$. Show that:

1-$FG$, and the line through $B$ perpendicular to $AG$, and the line through $C$ perpendicular to $DG$ are concurrent at $P_{bc}$.

2-If $ABCD$ be a cyclic quadrilateral, show that $P_{ab}P_{bc}P_{cd}P_{da}$ be a tangential quadrilateral

Problem 2: Let $ABCD$ be a tangential quadrilateral, let $A_1,B_1,C_1,D_1$ be midpoint of $AB,BC,CD,DA$ respectively. Let $A_2,B_2,C_2,D_2$ lie on $AB,BC,CD,DA$ respectively. Such that $A_1C_2, B_1D_2,C_1A_2,D_1B_2$ perpendicular to $CD,DA,AB,BC$ respectively. Then the quadrilateral bounded by $A_1C_2, B_1D_2,C_1A_2,D_1B_2$ is a tangential quadrilateral.

97-A generalization Zeeman-Gossard perpector

Let $ABC$ be a triangle, let a line $L$ meets the Euler line at $D$ and the line $L$ meets three sidelines $BC,CA,AB$ at $A_0,B_0,C_0$ respectively. Let $H_a, H_b,H_c$ be the orthocenter of three triangles $AB_0C_0, BC_0A_0, CA_0B_0$ respectively.  Let $G_a,G_b,G_c$ be the centroid of three triangles $AB_0C_0, BC_0A_0, CA_0B_0$ respectively. Let three points  $D_a,D_b,D_c$ lie on the Euler line of $AB_0C_0, BC_0A_0, CA_0B_0$ respectively such that:

\[ \frac{\overline{D_aH_a}}{\overline{DaGa}}=\frac{\overline{D_bH_b}}{\overline{D_bG_b}}=\frac{\overline{D_cH_c}}{\overline{D_cG_c}} \space=\frac{\overline{DH}}{\overline{DG}}=t  \]

 Let $L_a,L_b,L_c$ = three lines through $D_a,D_b,D_c$ and parallel to $BC,CA,AB$ respectively.

View in Geogebra

Problem 1: Then triangle bounded by three line $(L_a,L_b,L_c)$ are congruent
and homothety to triangle ABC. The homothetic center lie on the line L.
When $L$ is the Euler line or t = ∞ the problem is Zeeman-Gossard perpector

Problem 2: Newton lines of four quadrilateral $(AB,AC,L_a,L)$, $(BC,BA,L_b,L)$,
$(CA,CB,L_c,L)$, $(AB,BC,CA,L)$ also through the homothetic center in problem 1

Where if $L_i$ $(i=1,2,...n)$ be a line, define $(L_1,L_2,....,L_n)$ = Polygon bound by $L_1,L_2,....,L_n$

96-Ten point circle associated X4240 in ETC

Let $ABC$ be a triangle, Let $(W)$ be the circle through $X(3),X(110)$ and $X(4240)$ in $ETC$. I found 7 points also lie on $(W)$ as follows:

Let the Euler line of ABC meets the sidelines $BC,CA,AB$ at $A_0,B_0,C_0$. Six points  $X(3), X(110)$ of three triangles $AB_0C_0, BA_0C_0, CB_0A_0$ also lie on $(W)$. The circumcenter $X(3)$ of Paralogic triangle whose perpectrix is the Euler line of $ABC$ also lie on $(W)$. So $10$ points lie on this circle.

Center of this circle is midpoint of $X(3)$ of $ABC$ and $O'$